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  • Cited by 314
Publisher:
Cambridge University Press
Online publication date:
July 2014
Print publication year:
2014
Online ISBN:
9781139814782

Book description

Boolean functions are perhaps the most basic objects of study in theoretical computer science. They also arise in other areas of mathematics, including combinatorics, statistical physics, and mathematical social choice. The field of analysis of Boolean functions seeks to understand them via their Fourier transform and other analytic methods. This text gives a thorough overview of the field, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry. Each chapter includes a 'highlight application' such as Arrow's theorem from economics, the Goldreich–Levin algorithm from cryptography/learning theory, Håstad's NP-hardness of approximation results, and 'sharp threshold' theorems for random graph properties. The book includes roughly 450 exercises and can be used as the basis of a one-semester graduate course. It should appeal to advanced undergraduates, graduate students and researchers in computer science theory and related mathematical fields.

Reviews

'The applications of the ideas in this book are plentiful and diverse, and O’Donnell does an excellent job of leading the reader from one viewpoint to the next. I found it especially enjoyable to see theorems that I’m personally familiar with as a cryptographer, such as the Goldreich–Levin theorem, placed alongside other things I didn’t know as well, like Arrow’s theorem from social choice - with everything woven into a single, consistent story. I suspect other ‘fresh readers’ will similarly find parts of this book that they recognize, and others they don’t. The relationships exposed between these ideas should be of interest to everyone. Altogether, I highly recommend that you take a glance at Analysis of Boolean Functions.'

Daniel Apon Source: SIGACT News

'This 423-page book is a rich source of material presented in an attractive form. Each chapter highlights one main result which provides a focus and incentive for the reader to go to the end of the chapter.'

Martin C. Cooper Source: MathSciNet

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Contents

Bibliography
Aaronson, Scott (2008) How to solve longstanding open problems in quantum computing using only Fourier analysis. Lecture at Banff International Research Station. http://www.scottaaronson.com/talks/openqc.ppt.
Aaronson, Scott, and Andris, Ambainis (2011) The need for structure in quantum speedups. In: Proceedings of the 2nd Annual Innovations in Theoretical Computer Science Conference. pp. 338-352.
Achlioptas, Dimitris, and Ehud, Friedgut (1999) A sharp threshold for £-colorability. Random Structures & Algorithms, 14(1):63-70.
Achlioptas, Dimitris, and Assaf, Naor (2005) The two possible values of the chromatic number of arandom graph. Annals of Mathematics, 162(3):1335-1351.
Adams, Robert, and Frank, Clarke (1979) Gross's logarithmic Sobolev inequality: a simple proof. American Journal of Mathematics, 101(6):1265-1269.
Ajtai, Miklos (1983) Ej-formulae on finite structures. Annals of Pure and Applied Logic, 24(1):1-48.
Ajtai, Miklós, and Nathal, Linial (1993) The influence of large coalitions. Combinatorica, 13(2):129-145.
Alon, Noga, László, Babai, and Alon, Itai (1985) A fast and simple randomized algorithm for the maximal independent set problem. Journal of Algorithms, 7(4):567-583.
Alon, Noga, Oded, Goldreich, Johan, Hastad, and Rene, Peralta (1992) Simple constructions of almost £-wise independent random variables. Random Structures & Algorithms, 3(3):289-304.
Alon, Noga, and Joel, Spencer (2008) The Probabilistic Method. 3rd ed. Wiley-Interscience.
Alon, Noga, and Benjamin, Sudakov (2000) Bipartite subgraphs and the smallest eigenvalue. Combinatorics, Probability and Computing, 9(1):1-12.
Alon, Noga, Benny, Sudakov, and Uri, Zwick (2002) Constructing worst case instances for semidefinite programming based approximation algorithms. SIAM Journal on Discrete Mathematics, 15(1):58-72.
Amano, Kazuyuki (2011) Tight bounds on the average sensitivity of k-CNF. Theory of Computing, 7(1):45-48.
Ambainis, Andris (2003) Polynomial degree vs. quantum query complexity. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science. pp. 230-239.
Ambrosio, Luigi, and Alessio, Figalli (2011) Surface measures and convergence of the Ornstein-Uhlenbeck semigroup in Wiener spaces. Annales de la faculté des sciences de Toulouse Mathématiques (série 6), 20(2):407-438.
Ambrosio, Luigi, Alessio, Figalli, and Eris, Runa (2013) On sets of finite perimeter in Wiener spaces: reduced boundary and convergence to halfspaces. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni, 24(1):111-122.
Ambrosio, Luigi, Nicola, Fusco, and Diego, Pallara (2000) Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press.
Ambrosio, Luigi, Michele, Miranda Jr., Stefania, Maniglia, and Diego, Pallara (2010) BV functions in abstract Wiener spaces. Journal of Functional Analysis, 258(3):785-813.
Arora, Sanjeev, Eli, Berger, Elad, Hazan, Guy, Kindler, and Muli, Safra (2005) On non-approximability for quadratic programs. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science. pp. 206-215.
Arora, Sanjeev, Carsten, Lund, Rajeev, Motwani, Madhu, Sudan, and Mario, Szegedy (1998) Proof verification and the hardness of approximation problems. Journal of the ACM, 45(3):501-555.
Arora, Sanjeev, and Shmuel, Safra (1998) Probabilistic checking of proofs: a new characterization of NP. Journal of the ACM, 45(1):70-122.
Arrow, Kenneth (1950) A difficulty in the concept of social welfare. Journal of Political Economy, 58(4):328-346.
Arrow, Kenneth (1963) Social Choice and Individual Values. Cowles Foundation.
Austrin, Per (2007a) Balanced Max-2Sat might not be hardest. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing. pp. 189-197.
Austrin, Per (2007b) Towards sharp inapproximability for any 2-CSP. In: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science. pp. 307-317.
Baernstein, Albert, and Bert, Taylor (1976) Spherical rearrangements, subharmonic functions, and *-functions in n-space. Duke Mathematical Journal, 43(2):245-268.
Bakry, Dominique (1994) L'hypercontractivité et son utilisation en théorie des semi-groupes. In: Lectures on Probability Theory (Saint-Flour, 1992), Springer, Berlin, vol. 1581 of Lecture Notes in Mathematics. pp. 1-114.
Bakry, Dominique, and Michel, Ledoux (1996) Levy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator. Inventiones mathematicae, 123(1):259-281.
Bakry, Dominiques, and Michel, Émery (1985) Diffusions hypercontractives. In: Séminaire de Probabilites, XIX, Springer, Berlin, vol. 1123 of Lecture Notes in Mathematics. pp. 177-206.
Bal, Deepak (2013) On sharp thresholds of monotone properties: Bourgain's proof revisited. Tech. Rep. 1302.1162, arXiv.
Balashov, Leonid, and Aleksandr, Rubinshtein (1973) Series with respect to the Walsh system and their generalizations. Journal of Soviet Mathematics, 1(6):727-763.
Ball, Keith (1993) The reverse isoperimetric problem for gaussian measure. Discrete and Computational Geometry, 10(4):411-420.
Ball, Keith, Franck, Barthe, Witold, Bednorz, Krzysztof, Oleszkiewicz, and Pawel, Wolff (2013) L1-smoothing for the Ornstein-Uhlenbeck semigroup. Mathematika, 59(1):160-168.
Banzhaf, John (1965) Weighted voting doesn't work: a mathematical analysis. Rutgers Law Review, 19:317-343.
Barak, Boaz, Fernando, Brandão, Aram, Harrow, Jonathan, Kelner, David, Steurer, and Yuan, Zhou (2012) Hypercontractivity, sum-of-squares proofs, and their applications. In: Proceedings of the 44th Annual ACM Symposium on Theory of Computing. pp. 307-326.
Barbour, Andrew, and Peter, Hall (1984) Stein's method and the Berry-Esseen theorem. Australian Journal of Statistics, 26(1):8-15.
Barthe, Franck, and Bernard, Maurey (2000) Some remarks on isoperimetry of Gaussian type. Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 36(4):419-434.
Beame, Paul (1994) A switching lemma primer. Tech. Rep. UW-CSE-95-07-01, University of Washington.
Beckner, William (1975) Inequalities in Fourier analysis. Annals of Mathematics, 102:159-182.
Beckner, William (1992) Sobolev inequalities, the Poisson semigroup, and analysis on the sphere Sn. Proceedings of the National Academy of Sciences, 89(11):4816-4819.
Bellare, Mihir, Don, Coppersmith, Johan, Hastad, Marcos, Kiwi, and Madhu, Sudan (1996) Linearity testing in characteristic two. IEEE Transactions on Information Theory, 42(6):1781-1795.
Bellare, Mihir, Oded, Goldreich, and Madhu, Sudan (1995) Free bits, PCPs, and non-approximability - towards tight results. Tech. Rep. TR95-024, Electronic Colloquium on Computational Complexity.
Bellare, Mihir, Oded, Goldreich, and Madhu, Sudan (1998) Free bits, PCPs, and non-approximability - towards tight results. SIAM Journal of Computing, 27(3):804-915.
Ben-Aroya, Avraham, and Amnon, Ta-Shma (2009) Constructing small-bias sets from algebraic-geometric codes. In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science. pp. 191-197.
Ben-Or, Michael, and Nathan, Linial (1985) Collective coin flipping, robust voting schemes and minima of Banzhaf values. In: Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science. pp. 408-416.
Ben-Or, Michael, and Nathan, Linial (1990) Collective coin flipping. In: Randomness and Computation, ed. Silvio, Micali and Franco, Preparata, JAI Press, vol. 5 of Advances in Computing Research: A Research Annual. pp. 91-115.
Ben-Sasson, Eli, Oded, Goldreich, Prahladh, Harsha, Madhu, Sudan, and Salil, Vadhan (2004) Robust PCPs of proximity, shorter PCPs and applications to coding. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing. pp. 1-10.
Ben-Sasson, Eli, and Madhu, Sudan (2008) Short PCPs with polylog query complexity. SIAM Journal on Computing, 38(2):551-607.
Ben-Sasson, Eli, Madhu, Sudan, Salil, Vadhan, and Avi, Wigderson (2003) Randomness-efficient low degree tests and short PCPs via epsilon-biased sets. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing. pp. 612-621.
Benjamini, Itai, Gil, Kalai, and Oded, Schramm (1999) Noise sensitivity of Boolean functions and applications to percolation. Publications Mathematiques de l'IHÉS, 90(1):5-43.
Benjamini, Itai, Oded, Schramm, and David, Wilson (2005) Balanced Boolean functions that can be evaluated so that every input bit is unlikely to be read. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing. pp. 244-250.
Bentkus, Vidmantas (2004) A Lyapunov type bound in Rd. Rossiĭskaya Akademiya Nauk. Teoriya Veroyatnosteĭ i ee Primeneniya, 49(2):400-410.
Benyamini, Yoav (1984) Two-point symmetrization, the isoperimetric inequality on the sphere and some applications. In: Texas Functional Analysis Seminar, 1983-1984, vol. 1984. pp. 53-76.
Bernasconi, Anna, and Bruno, Codenotti (1999) Spectral analysis of Boolean functions as a graph eigenvalue problem. IEEE Transactions on Computers, 48(3):345-351.
Berry, Andrew (1941) The accuracy of the Gaussian approximation to the sum of independent variates. Transactions of the American Mathematical Society, 49(1):122-139.
Bikelis, Algimantas (1966) Estimates of the remainder in a combinatorial central limit theorem. Litovskii Matematicheskii Sbornik, 6(3):323-346.
Blais, Eric, Ryan, O'Donnell, and Karl, Wimmer (2010) Polynomial regression under arbitrary product distributions. Machine Learning, 80(2):273-294.
Blau, Julian (1957) The existence of social welfare functions. Econometrica, 25(2):302-313.
Blum, Avrim (2003) Learning a function of rrelevant variabless. In: Proceedings of the 16th Annual Conference on Learning Theory, ed. Bernhard, Schölkopf and Manfred, Warmuth. Springer, Berlin, vol. 2777 of Lecture Notes in Computer Science. pp. 731-733.
Blum, Manuel, and Russell, Impagliazzo (1987) Generic oracles and oracle classes. In: Proceedings of the 28th Annual IEEE Symposium on Foundations ofComputer Science. pp. 118-126.
Blum, Manuel, Michael, Luby, and Ronitt, Rubinfeld (1990) Self-testing/correcting with applications to numerical problems. In: Proceedings of the 22nd Annual ACM Symposium on Theory of Computing. pp. 73-83.
Blumer, Anselm, Andrzej, Ehrenfeucht, David, Haussler, and Manfred, Warmuth (1987) Occam's razor. Information Processing Letters, 24(6):377-380.
Bobkov, Sergey (1997) An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space. Annals of Probability, 25(1):206-214.
Bobkov, Sergey, and Friedrich, Götze (1999) Discrete isoperimetric and Poincare-type inequalities. Probability Theory and Related Fields, 114(2):245-277.
Bobkov, Sergey, and Michel, Ledoux (1998) On modified logarithmic Sobolev Inequalities for Bernoulli and Poisson measures. Journal of Functional Analysis, 156(2):347-365.
Bogachev, Vladimir (1998) Gaussian Measures. Mathematical Series and Monographs.
American Mathematical Society. Bogdanov, Andrej, and Emanuele, Viola (2007) Pseudorandom bits for polynomials. In: Proceedings of the 48th Annual IEEE Symposium on Foundations ofComputer Science. pp. 41-51.
Bollobás, Béla (2001) Random Graphs. Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge.
Bollobás, Béla, and Oliver, Riordan (2008) Random graphs and branching processes. In: Handbook of Large-Scale Random Networks, ed. Béla, Bollobás, Robert, Kozma, and Dezsó, Miklós, Springer. pp. 15-116.
Bollobás, Béla, and Andrew, Thomason (1987) Threshold functions. Combinatorica, 7(1):35-38.
Bolthausen, Erwin (1984) An estimate of the remainder in a combinatorial central limit theorem. Probability Theory and Related Fields, 66(3):379-386.
Bonami, Aline (1968) Ensembles ∧(p) dans le dual de D∞. Annales de l'Institut Fourier, 18(2):193-204.
Bonami, Aline (1970) Étude des coefficients Fourier des fonctions de Lp(G). Annales de l'Institut Fourier, 20(2):335-402.
Boppana, Ravi (1997) The average sensitivity of bounded-depth circuits. Information Processing Letters, 63(5):257-261.
Borell, Christer (1975) The Brunn-Minkowski inequality in Gauss space. Inventiones Mathematicae, 30(2):207-216.
Borell, Christer (1979) On the integrability of Banach space valued Walsh polynomials. In: Séminaire de Probabilites, XIII, Springer, Berlin, vol. 721 of Lecture Notes in Mathematics. pp. 1-3.
Borell, Christer (1982) Positivity improving operators and hypercontractivity. Mathematische Zeitschrift, 180(2):225-234.
Borell, Christer (1984) On polynomial chaos and integrability. Probabability and Mathematical Statistics, 3(2):191-203.
Borell, Christer (1985) Geometric bounds on the Ornstein–Uhlenbeck velocity process. Probability Theory and Related Fields, 70(1):1-13.
Bourgain, Jean (1979) Walsh subspaces of Lpproduct spaces. In: Séminaire D'Analyse Fonctionnelle, École Polytechnique, Centre De Mathematiques, pp. IV.1-IV.9.
Bourgain, Jean (1999) On sharp thresholds of monotone properties. Journal of the American Mathematical Society, 12(4):1046-1053. Appendix to the main paper, Sharp thresholds of graph properties, and the k-sat problem by Ehud Friedgut.
Bourgain, Jean, Jeff, Kahn, Gil, Kalai, Yitzhak, Katznelson, and Nathan, Linial (1992) The influence of variables in product spaces. Israel Journal of Mathematics, 77(1):55-64.
Brams, Steven, William, Gehrlein, and Fred, Roberts, eds. (2009) The Mathematics of Preference, Choice and Order. Springer.
Brandman, Yigal (1987) Spectral lower-bound techniques for logic circuits. Ph.D. thesis, Stanford University.
Brandman, Yigal, Alon, Orlitsky, and John, Hennessy (1990) A spectral lower bound technique for the size of decision trees and two-level AND/OR circuits. IEEE Transactions on Computers, 39(2):282-287.
Brascamp, Herm, and Elliott, Lieb (1976) Best constants in Young's inequality, its converse, and its generalization to more than three functions. Advances in Mathematics, 20(2):151-173.
Broadbent, Simon, and John, Hammersley (1957) Percolation processes I. Crystals and mazes. Mathematical Proceedings of the Cambridge Philosophical Society, 53(3):629-641.
Bruck, Jehoshua (1990) Harmonic analysis of polynomial threshold functions. SIAM Journal on Discrete Mathematics, 3(2):168-177.
Bruck, Jehoshua, and Roman, Smolensky (1992) Polynomial threshold functions, AC0 functions and spectral norms. SIAM Journal on Computing, 21(1):33-42.
Bshouty, Nader, and Christino, Tamon (1996) On the Fourier spectrum of monotone functions. Journal of the ACM, 43(4):747-770.
Capitaine, Mireille, Elton, Hsu, and Michel, Ledoux (1997) Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces. Electronic Communications in Probability, 2:71-81.
Carbery, Anthony, and James, Wright (2001) Distributional and Lq norm inequalities for polynomials over convex bodies in Rn. Mathematical Research Letters, 8(3):233-248.
Carlen, Eric, and Michael, Loss (1990) Extremals of functionals with competing symmetries. Journal of Functional Analysis, 88(2):437-456.
Carlet, Claude (2010) Boolean functions for cryptography and error-correcting codes. In: Boolean Models and Methods in Mathematics, Computer Science, and Engineering, ed. Yves, Crama and Peter, Hammer, Cambridge University Press. pp. 257-397.
Charikar, Moses, and Anthony, Wirth (2004) Maximizing quadratic programs: extending Grothendieck's Inequality. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science. pp. 54-60.
Chebyshev, Pafnuty (1860) Sur le developpement des fonctions a une seule variable. Bulletin de l'Academie imperiale des sciences de St.-Petersbourg, 1:193-200.
Chen, Louis, Larry, Goldstein, and Qi-Man, Shao (2011) Normal Approximation by Stein's Method. Springer.
Chor, Benny, Joel, Friedmann, Oded, Goldreich, Johan, Hastad, Steven, Rudich, and Roman, Smolensky (1985) The bit extraction problem or t-resilient functions. In: Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science. pp. 396-407.
Chor, Benny, and Mihaly, Gereb-Graus (1987) On the influence of single participant in coin flipping schemes. Tech. rep., Harvard University.
Chor, Benny, and Mihaly, Gereb-Graus (1988) On the influence of single participant in coin flipping schemes. SIAM Journal on Discrete Mathematics, 1(4):411-415.
Chow, Chao-Kong (1961) On the characterization of threshold functions. In: Proceedings of the 2nd Annual Symposium on Switching Circuit Theory and Logical Design (FOCS). pp. 34-38.
Chung, Fan, and Ronald, Graham (1992) Quasi-random subsets of Zn. Journal of Combinatorial Theory, Series A, 61:64-86.
Chung, Fan, Ronald, Graham, and Richard, Wilson (1989) Quasi-random graphs. Com- binatorica, 9(4):345-362.
Coleman, John (1971) Control of collectivities and the power of a collectivity to act. In: Social Choice, ed. Bernhardt Lieberman. Gordon and Breach.
Creignou, Nadia, Sanjeev, Khanna, and Madhu, Sudan (2001) Complexity Classifications ofBoolean Constraint Satisfaction Problems. Society for Industrial and Applied Mathematics.
De, Anindya, Elchanan, Mossel, and Joe, Neeman (2012) Majority is Stablest: discrete andSoS. Tech. Rep. 1211.1001, arXiv.
De, Anindya, Elchanan, Mossel, and Joe, Neeman (2013) Majority is Stablest: Discrete and SoS. In: Proceedings of the 45th Annual ACM Symposium on Theory of Computing.
de Condorcet, Nicolas (1785) Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix. Paris, de l'imprimerie royale.
de Klerk, Etienne, Dmitrii, Pasechnik, and Johannes, Warners (2004) On approximate graph colouring and MAX-k-CUT algorithms based on the ϑ-function. Journal of Combinatorial Optimization, 8(3):267-294.
Delorme, Charles, and Svatopluk, Poljak (1993) Laplacian eigenvalues and the maximum cut problem. Mathematical Programming, 62(1-3):557-574.
Diaconis, Persi, and David, Freedman (1987) A dozen de Finetti-style results in search of a theorem. Annales de l'Institut Henri Poincaré (B), 23(S2):397-423.
Diaconis, Persi, and Laurent, Saloff-Coste (1996) Logarithmic Sobolev inequalities for finite Markov chains. Annals of Applied Probability, 6(3):695-750.
Diakonikolas, Ilias, Prahladh, Harsha, Adam, Klivans, Raghu, Meka, Prasad, Raghavendra, Rocco, Servedio, and Li-Yang, Tan (2010) Bounding the average sensitivity and noise sensitivity of polynomial threshold functions. In: Proceedings of the 42nd Annual ACM Symposium on Theory of Computing. pp. 533-542.
Diakonikolas, Ilias, and Rocco, Servedio (2009) Improved approximation of linear threshold functions. In: Proceedings of the 24th Annual IEEE Conference on Computational Complexity. pp. 161-172.
Dickson, Leonard (1901) Linear Groups with an Exposition of Galois Field Theory. B. G. Teubner.
Dillon, John (1972) A survey of bent functions. NSA Technical Journal, 191-215.
Dinur, Irit (2007) The PCP Theorem by gap amplification. Journal of the ACM, 54(3):1-44.
Dinur, Irit, Ehud, Friedgut, Guy, Kindler, and Ryan, O'Donnell (2007) On the Fourier tails of bounded functions over the discrete cube. Israel Journal of Mathematics, 160(1):389-U2.
Dinur, Irit, and Omer, Reingold (2004) Assignment testers: towards a combinatorial proof of the PCP theorem. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science. pp. 155-164.
Efron, Bradley, and Charles, Stein (1981) The jackknife estimate of variance. Annals of Statistics, 9(3):586-596.
Ehrhard, Antoine (1983) Symétrisation dans l'espace de Gauss. Mathematica Scandinavia, 53:281-301.
Ehrhard, Antoine (1984) Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes. Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 17(2):317-332.
Einstein, Albert (1905) Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik, 322(8):549-560.
Eldan, Ronen (2013) A two-sided estimate for the Gaussian noise stability deficit.Tech. Rep. 1307.2781, arXiv.
Elgot, Calvin (1961) Truth functions realizable by single threshold organs. In: Proceedings of the 2nd Annual Symposium on Switching Circuit Theory and Logical Design (FOCS). pp. 225-245.
Enflo, Per (1970) On the nonexistence of uniform homeomorphisms between Lp-spaces. Arkiv för Matematik, 8(2):103-105.
Epperson, Jay (1989) The hypercontractive approach to exactly bounding an operator with complex Gaussian kernel. Journal of Functional Analysis, 87(1):1-30.
Erdos, Paul, and Alfréd, Rényi (1959) On random graphs I. Publicationes Mathematicae Debrecen, 6:290-297.
Ergün, Funda, Ravi, Kumar, and Ronitt, Rubinfeld (1999) Fast approximate PCPs. In: Proceedings of the 31st Annual ACM Symposium on Theory of Computing. pp. 41-50.
Esseen, Carl-Gustav (1942) On the Liapounoff limit of error in the theory of probability. Arkiv för matematik, astronomi och fysik, 28(9):1-19.
Falik, Dvir, and Alex, Samorodnitsky (2007) Edge-isoperimetric inequalities and influences. Combinatorics, Probability and Computing, 16(5):693-712.
Federbush, Paul (1969) Partially alternate derivation of a result of Nelson. Journal of Mathematical Physics, 10:50-52.
Feige, Uriel, Shafi, Goldwasser, Lászlo, Lovász, Shmuel, Safra, and Mario, Szegedy (1996) Interactive proofs and the hardness of approximating cliques. Journal of the ACM, 43(2):268-292.
Feige, Uriel, and Michael, Langberg (2006) The RPR2 rounding technique for semidefinite programs. Journal of Algorithms, 60(1):1-23.
Feige, Uriel, and Lászlo, Lovász (1992) Two-prover one-round proof systems, their power and their problems. In: Proceedings of the 24th Annual ACM Symposium on Theory of Computing. pp. 733-744.
Feige, Uriel, and Gideon, Schechtman (2002) On the optimality of the random hyperplane rounding technique for Max-Cut. Randoom Structures and Algorithms, 20(3):403-440.
Figiel, Tadeusz, Joram, Lindenstrauss, and Vitali, Milman (1977) The dimension of almost spherical sections of convex bodies. Acta Mathematica, 139(1-2):53-94.
Fine, Nathan (1949) On the Walsh functions. Transactions of the American Mathematical Society, 65(3):372-414.
Freivalds, Rūsinš (1979) Fast probabilistic algorithms. In: Proceedings of the 4th Annual International Symposium on Mathematical Foundations of Computer Science. pp. 57-69.
Friedgut, Ehud (1998) Boolean functions with low average sensitivity depend on Few coordinates. Combinatorica, 18(1):27-36.
Friedgut, Ehud (1999) Sharp thresholds of graph properties, and the k-SAT problem. Journal of the American Mathematical Society, 12(4):1017-1054.
Friedgut, Ehud (2005) Hunting for sharp thresholds. Random Structures & Algorithms, 26(1-2):37-51.
Friedgut, Ehud, and Gil, Kalai (1996) Every monotone graph property has a sharp threshold. Proceedings of the American Mathematical Society, 124(10):2993-3002.
Friedgut, Ehud, Gil, Kalai, and Assaf, Naor (2002) Boolean functions whose Fourier transform is concentrated on the first two levels and neutral social choice. Advances in Applied Mathematics, 29(3):427-437.
Friedl, Katalin, and Madhu, Sudan (1995) Some improvements to total degree tests. In: Proceedings of the 3rd Annual Israel Symposium on Theory of Computing Systems. pp. 190-198.
Furst, Merrick, Jeffrey, Jackson, and Sean, Smith (1991) Improved learning of AC0 functions. In: Proceedings of the 4th Annual Conference on Learning Theory. pp. 317-325.
Furst, Merrick, James, Saxe, and Michael, Sipser (1984) Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 17(1):13-27.
Garman, Mark, and Morton, Kamien (1968) The paradox of voting: probability calculations. Behavioral Science, 13(4):306-316.
Girko, Vyacheslav (1973) Limit theorems for random quadratic forms. Izdat. Naukova Dumka, pp. 14-30.
Glimm, James (1968) Boson fields with nonlinear selfinteraction in two dimensions. Communications in Mathematical Physics, 8(1):12-25.
Goemans, Michel, and David, Williamson (1994) A 0.878 approximation algorithm for MAX-2SAT and MAX-CUT. In: Proceedings of the 26th Annual ACM Symposium on Theory of Computing. pp. 422-431.
Goemans, Michel, and David, Williamson (1995) Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM, 42:1115-1145.
Goldmann, Mikael, and Alexander, Russell (2000) Spectral bounds on general hard core predicates. In: Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science. pp. 614-625.
Goldreich, Oded, Shafi, Goldwasser, and Dana, Ron (1998) Property testing and its connections to learning and approximation. Journal of the ACM, 45(4):653-750.
Goldreich, Oded, and Leonid, Levin (1989) A hard-core predicate for all one-way functions. In: Proceedings of the 21st Annual ACM Symposium on Theory of Computing. pp. 25-32.
Golomb, Solomon (1959) On the classification of Boolean functions. IRE Transactions on Circuit Theory, 6(5):176-186.
Gopalan, Parikshit, Adam, Kalai, and Adam, Klivans (2008) Agnostically learning decision trees. In: Proceedings of the 40th Annual ACM Symposium on Theory of Com-puting. pp. 527-536.
Gopalan, Parikshit, Raghu, Meka, and Omer, Reingold (2012) DNF Sparsification and a faster deterministic counting algorithm. In: Proceedings of the 26th Annual IEEE Conference on Computational Complexity. pp. 126-135.
Gopalan, Parikshit, Ryan, O'Donnell, Rocco, Servedio, Amir, Shpilka, and Karl, Wimmer (2011) Testing Fourier dimensionality and sparsity. SIAM Journal on Computing, 40(4):1075-1100.
Gopalan, Parikshit, Ryan, O'Donnell, Yi, Wu, and David, Zuckerman (2010) Fooling functions of halfspaces under product distributions. In: Proceedings of the 25th Annual IEEE Conference on Computational Complexity. pp. 223-234.
Gotsman, Craig, and Nathan, Linial (1994) Spectral properties of threshold functions. Combinatorica, 14(1):35-50.
Gowers, W. Timothy (2001) A new proof of Szemeredi's theorem. Geometric and Functional Analysis, 11(3):465-588.
Green, Ben, and Tom, Sanders (2008) Boolean functions with small spectral norm. Geometric and Functional Analysis, 18(1):144-162.
Gross, Leonard (1972) Existence and uniqueness of physical ground states. Journal of Functional Analysis, 10:52-109.
Gross, Leonard (1975) Logarithmic Sobolev inequalities. American Journal of Mathematics, 97(4):1061-1083.
Guilbaud, George-Théodule (1952) Les théories de l'intérêt général et le problème logique de l'agrégation. Economie appliquée, V(4):501-551.
Haagerup, Uffe (1982) The best constants in the Khinchine Inequality. Studia Mathematica, 70(3):231-283.
Haar, Alfréd (1910) Zur Theorie der orthogonalen Funktionensysteme. Mathematische Annalen, 69(3):331-371.
Hajek, Jaroslav (1968) Asymptotic normality of simple linear rank statistics under alternatives. Annals of Mathematical Statistics, 39(2):325-346.
Harper, Lawrence (1964) Optimal assignments of numbers to vertices. Journal of the Society for Industrial and Applied Mathematics, 12(1):131-135.
Harsha, Prahladh, Adam, Klivans, and Raghu, Meka (2010) Bounding the sensitivity of polynomial threshold functions. In: Proceedings of the 42nd Annual ACM Symposium on Theory of Computing. pp. 533-542.
Håstad, Johan (1987) Computational Limitations for Small Depth Circuits. MIT Press.
Håstad, Johan (1996) Testing of the long code and hardness for clique. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing. pp. 11-19.
Håstad, Johan (1997) Some optimal inapproximability results. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing. pp. 1-10.
Håstad, Johan (1999) Clique is hard to approximate within n1–ε. Acta Mathematica, 182(1):105-142.
Håstad, Johan (2001b) Some optimal inapproximability results. Journal of the ACM, 48(4):798-859.
Håstad, Johan (2012) On the correlation of parity and small-depth circuits. Tech. Rep. TR12-137, Electronic Colloquium on Computational Complexity.
Hatami, Hamed (2012) A structure theorem for Boolean functions with small total influences. Annals of Mathematics, 176(1):509-533.
Hermite, Charles (1864) Sur un nouveau développement en série des fonctions. Comptes rendus de l'Académie des sciences, 58(2):93-100, 266-273.
Higuchi, Yasunari, and Nobuo, Yoshida (1995) Analytic conditions and phase transition for Ising models. Unpublished lecture notes (in Japanese).
Hino, Masanori (2010) Sets of finite perimeter and the Hausdorff-Gauss measure on the Wiener space. Journal of Functional Analysis, 258(5):1656-1681.
Hoeffding, Wassily (1948) A class of statistics with asymptotically normal distribution. Annals ofMathematical Statistics, 19(3):293-325.
Hurst, Stanley, D. Michael, Miller, and Jon, Muzio (1982) Spectral method of Boolean function complexity. Electronics Letters, 18(13):572-574.
Impagliazzo, Russell, William, Matthews, and Ramamohan, Paturi (2012) A satisfiability algorithm for AC0.In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms. pp. 961-972.
Jackson, Jeffrey (1995) The Harmonic Sieve: a novel application of Fourier analysis to machine learning theory and practice. Ph.D. thesis, Carnegie Mellon University.
Jackson, Jeffrey (1997) An efficient membership-query algorithm for learning DNF with respect to the uniform distribution. Journal of Computer and System Sciences, 55(3):414-440.
Jain, Rahul, and Shengyu, Zhang (2011) The influence lower bound via query elimination. Theory of Computing, 7(1):147-153.
Janson, Svante (1997) Gaussian Hilbert Spaces. Cambridge University Press.
Jayram, T. S., Ravi, Kumar, and D., Sivakumar (2003) Two applications of information complexity. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing. pp. 673-682.
Jendrej, Jacek, Krzysztof, Oleszkiewicz, and Jakub, Wojtaszczyk (2012) On some extensions of the FKN theorem. Manuscript.
Johnson, David (1974) Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences, 9(3):256-278.
Kahane, Jean-Pierre (1968) Some Random Series of Functions. D. C. Heath & Co.
Kahn, Jeff, and Gil, Kalai (2007) Thresholds and expectation thresholds. Combinatorics, Probability and Computing, 16(3):495-502.
Kahn, Jeff, Gil, Kalai, and Nathan, Linial (1988) The influence of variables on Boolean functions. In: Proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science. pp. 68-80.
Kalai, Gil (2002) A Fourier-theoretic perspective on the Condorcet paradox and Arrow's theorem. Advances in Applied Mathematics, 29(3):412-426.
Kane, Daniel (2011) On Elliptic Curves, the ABC Conjecture, and Polynomial Threshold Functions. Ph.D. thesis, Harvard University.
Kane, Daniel (2012) The Correct Exponent for the Gotsman-Linial Conjecture. Tech. Rep. 1210.1283, arXiv.
Karlin, Samuel, and Yosef, Rinott (1982) Applications of ANOVA type decompositions for comparisons of conditional variance statistics including jackknife estimates. Annals of Statistics, 10(2):485-501.
Karloff, Howard (1999) How good is the Goemans-Williamson MAX CUT algorithm?SIAM Journal of Computing, 29(1):336-350.
Karloff, Howard, and Uri, Zwick (1997) A 7/8-approximation algorithm for MAX 3SAT? In: Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science. pp. 406-415.
Karpovsky, Mark (1976) Finite Orthogonal Series in the Design of Digital Devices: Analysis, Synthesis, and Optimization. Wiley.
Kauers, Manuel, Ryan, O'Donnell, Li-Yang, Tan, and Yuan, Zhou (2013) Hypercontractive inequalities via SOS, with an application to Vertex-Cover. Manuscript.
Kaufman, Tali, Simon, Litsyn, and Ning, Xie (2010) Breaking the e-soundness bound of the linearity test over GF(2). SIAM Journal on Computing, 39(5):1988-2003.
Khanna, Sanjeev, Madhu, Sudan, Luca, Trevisan, and David, Williamson (2001) The approximability of constraint satisfaction problems. SIAM Journal on Computing, 30(6):1863-1920.
Kharitonov, Michael (1993) Cryptographic hardness of distribution-specific learning. In: Proceedings of the 25th Annual ACM Symposium on Theory of Computing. pp. 372-381.
Khot, Subhash (2002) On the power of unique 2-prover 1 -round games. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing. pp. 767-775.
Khot, Subhash (2005) Inapproximability results via Long Code based PCPs. ACM SIGACT News, 36(2):25-42.
Khot, Subhash (2010a) Inapproximability of NP-complete problems, discrete Fourier analysis, and geometry. In: Proceedings of the International Congress of Mathemati-cians. vol. 901, pp. 2676-2697.
Khot, Subhash (2010b) On the Unique Games Conjecture. In: Proceedings of the 25th Annual IEEE Conference on Computational Complexity. pp. 99-121.
Khot, Subhash, Guy, Kindler, Elchanan, Mossel, and Ryan, O'Donnell (2004) Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science. pp. 146-154.
Khot, Subhash, Guy, Kindler, Elchanan, Mossel, and Ryan, O'Donnell (2007) Optimal inapproximability results for Max-Cut and other 2-variable CSPs?SIAM Journal on Computing, 37(1):319-357.
Khot, Subhash, and Ryan, O'Donnell (2006) SDP gaps and UGC-hardness for Max-Cut-Gain. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science. pp. 217-226.
Khot, Subhash, and Oded, Regev (2008) Vertex cover might be hard to approximate to within 2 — ε. Journal of Computer and System Sciences, 74(3):335-349.
Khot, Subhash, and Nisheeth, Vishnoi (2005) The Unique Games Conjecture, integrality gap for cut problems and embeddability of negative type metrics into l1. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science. pp. 53-62.
Kiener, Konrad (1969) Über Produkte von quadratisch integrierbaren Funktionen endlicher Vielfalt. Ph.D. thesis, Universität Innsbruck.
Kindler, Guy (2002) Property testing, PCP, and juntas. Ph.D. thesis, Tel Aviv University.
Kindler, Guy, and Ryan, O'Donnell (2012) Gaussian noise sensitivity and Fourier tails. In: Proceedings of the 26th Annual IEEE Conference on Computational Complexity. pp. 137-147.
Kindler, Guy, and Shmuel, Safra (2002) Noise-resistant Boolean functions are juntas. Manuscript.
Kleitman, Daniel (1966) Families of non-disjoint subsets. Journal of Combinatorial Theory, 1(1):153-155.
Klivans, Adam, Ryan, O'Donnell, and Rocco, Servedio (2004) Learning intersections and thresholds of halfspaces. Journal of Computer and System Sciences, 68(4):808-840.
Klivans, Adam, Ryan, O'Donnell, and Rocco, Servedio (2008) Learning geometric concepts via Gaussian surface area. In: Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science. pp. 541-550.
Krakowiak, Wieslaw, and Jerzy, Szulga (1988) Hypercontraction principle and random multilinear forms. Probability Theory and Related Fields, 77(3):325-342.
Krause, Matthias, and Pavel, Pudlak (1997) On the computational power of depth-2 circuits with threshold and modulo gates. Theoretical Computer Science, 174(1-2):137-156.
Kravchuk, Mikahil (Krawtchouk) (1929) Sur une generalisation des polynomes d'Hermite. Comptes rendus de l'Académie des sciences, 189:620-622.
Kushilevitz, Eyal, and Yishay, Mansour (1993) Learning decision trees using the Fourier spectrum. SIAM Journal on Computing, 22(6):1331-1348.
Kwapień, Stanisław (2010) On Hoeffding decomposition in Lp. Illinois Journal of Mathematics, 54(3):1205-1211.
Kwapień, Stanisław, and Wojbor, Woyczyński (1992) Random Series and Stochastic Integrals: Single and Multiple. Probability and Its Applications. Birkhäusen.
Lamport, Leslie, Robert, Shostak, and Marshall, Pease (1982) The Byzantine generals problem. ACM Transactions on Programming Languages and Systems, 4(3):382-401.
Laplace, Pierre-Simon (1811) Mémoire sur les intégrales définies et leur application aux probabilités, et spécialement à la recherche du milieu qu'il faut choisir entre les résultats des observations. Mémoires de la Classe des Sciences Mathématiques et Physiques de l'Institut Impérial de France, Année 1810, 58:279-347.
Laplante, Sophie, Troy, Lee, and Mario, Szegedy (2006) The quantum adversary method and classical formula size lower bounds. Computational Complexity, 15(2):163-196.
Latala, Rafał, and Krzysztof, Oleszkiewicz (1994) On the best constant in the Khintchine–Kahane inequality. Studia Mathematica, 109(1):101-104.
Lebedev, Nikolai (1972) Special functions & their applications. Dover Publications.
Lechner, Robert (1963) Affine equivalence of switching functions. Ph.D. thesis, Harvard University.
Lechner, Robert (1971) Harmonic analysis of switching functions. In: Recent Developments in Switching Theory, ed. Amar, Mukhophadhay, Academic Press. pp. 121-228.
Ledoux, Michel (1992) On an integral criterion for hypercontractivity of diffusion semigroups and extremal functions. Journal of Functional Analysis, 105(2):444-465.
Ledoux, Michel (1994) Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space. Bulletin des Sciences Mathématiques, 118(6):485-510.
Ledoux, Michel (1996) Isoperimetry and Gaussian analysis. In: Lectures on Probability Theory and Statistics, ed. Pierre, Bernard, Springer, vol. 24 of Lecture Notes in Mathematics 1648. pp. 165-294.
Ledoux, Michel (1998) A short proof of the Gaussian isoperimetric inequality. In: High Dimensional Probability (Oberwolfach, 1996), Birkhauser, Basel, vol. 43 of Progress in Probability. pp. 229-232.
Ledoux, Michel (2013) Remarks on noise sensitivity, Brascamp-Lieb and Slepian inequalities. http://perso.math.univ-toulouse.fr/ledoux/files/2013/11/noise.pdf.
Lee, Homin (2010) Decision trees and influence: an inductive proof of the OSSS inequality. Theory of Computing, 6(1):81-84.
Leonardos, Nikos (2012) An improved lower bound for the randomized decision tree complexity ofrecursive majority. Tech. Rep. TR12-099, Electronic Colloquium on Computational Complexity.
Lévy, Paul (1922) Leçons d'Analyse Fonctionnelle. Gauthier-Villars.
Lindeberg, Jarl (1922) Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung. Mathematische Zeitschrift, 15(1):211-225.
Linial, Nathan, Yishay, Mansour, and Noam, Nisan (1989) Constant depth circuits, Fourier transform and Learnability. In: Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science. pp. 574-579.
Linial, Nathan, Yishay, Mansour, and Noam, Nisan (1993) Constant depth circuits, Fourier transform and learnability. Journal of the ACM, 40(3):607-620.
Lovett, Shachar (2008) Unconditional pseudorandom generators for low degree polynomials. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing. pp. 557-562.
Lovett, Shachar, and Yoav, Tzur (2009) Explicit lower bound for fooling polynomials by the sum of small-bias generators. In: Electronic Colloquium on Computational Complexity TR09-088.
Luby, Michael, Boban, Veličković, and Avi, Wigderson (1993) Deterministic approximate counting of depth-2 circuits. In: Proceedings of the 2nd Annual Israel Symposium on Theory of Computing Systems. pp. 18-24.
MacWilliams, F. Jessie, and Neil, Sloane (1977) The Theory of Error-Correcting Codes. North-Holland.
Magniez, Frédéric, Ashwin, Nayak, Miklos, Santha, and David, Xiao (2011) Improved bounds for the randomized decision tree complexity of recursive majority. In: Proceedings of the 38th Annual International Colloquium on Automata, Languages and Programming. pp. 317-329.
Mansour, Yishay (1994) Learning Boolean functions via the Fourier transform. In: Theoretical Advances in Neural Computation and Learning, ed. Vwani, Roychowd-hury, Kai-Yeung, Siu, and Alon, Orlitsky, Kluwer Academic Publishers, chap. 11. pp. 391-424.
Mansour, Yishay (1995) An 0(nlog log n) learning algorithm for DNF under the uniform distribution. Journal of Computer and System Sciences, 50(3):543-550.
Margulis, Grigory (1974) Probabilistic characteristics of graphs with large connectivity. Problemy Peredači Informacii, 10(2):101-108.
Matolcsi, Tamás, and József, Szücs (1973) Intersection des mesures spectrales conjuguees. Comptes rendus de l'Académie des sciences, 277:841-843.
Matthews, Gretchen, and Justin, Peachey (2011) Small-bias sets from extended norm-trace codes. Manuscript.
Matulef, Kevin, Ryan, O'Donnell, Ronitt, Rubinfeld, and Rocco, Servedio (2010) Testing halfspaces. SIAM Journal on Computing, 39(5):2004-2047.
May, Kenneth (1952) A set of independent necessary and sufficient conditions for simple majority decisions. Econometrica, 20(4):680-684.
McEliece, Robert, Eugene, Rodemich, Howard, Rumsey, and Lloyd, Welch (1977) New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities. IEEE Transactions on Information Theory, 23(2):157-166.
McKean, Henry (1973) Geometry of differential space. Annals of Probability, 1(2):197-206.
Mehler, F. Gustav (1866) Ueber die Entwicklung einer Function von beliebig vielen Variablen nach Laplaceschen Functionen höherer Ordnung. Journal fúr die reine und angewandte Mathematik, 66:161-176.
Midrijānis, Gatis (2004) Exact quantum query complexity for total Boolean functions. Quant-ph/0403168, arXiv.
Miranda, Michele Jr., (October 2013) Personal communication to the author.
Miranda, Michele Jr., Matteo, Novaga, and Diego, Pallara (2012) An introduction to BV functions in Wiener spaces. Tech. Rep. 1212.5926, arXiv.
Miranda, Michele, Diego, Pallara, Fabio, Paronetto, and Marc, Preunkert (2007) Short-time heat flow and functions of bounded variation in RN. Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série 6, 16(1):125-145.
Mossel, Elchanan (2010) Gaussian bounds for noise correlation offunctions. Geometric and Functional Analysis, 19(6):1713-1756.
Mossel, Elchanan, and Joe, Neeman (2012) Robust optimality of Gaussian noise stability. Tech. Rep. 1210.4126, arXiv.
Mossel, Elchanan, and Ryan, O'Donnell (2005) Coin flipping from a cosmic source: on error correction of truly random bits. Random Structures & Algorithms, 26(4):418-436.
Mossel, Elchanan, Ryan, O'Donnell, and Krzysztof, Oleszkiewicz (2005a) Noise stability of functions with low influences: invariance and optimality. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science. pp. 21-30.
Mossel, Elchanan, Ryan, O'Donnell, and Krzysztof, Oleszkiewicz (2005b) Noise stability of functions with low influences: invariance and optimality. Tech. Rep. math/0503503, arXiv.
Mossel, Elchanan, Ryan, O'Donnell, and Krzysztof, Oleszkiewicz (2010) Noise stability of functions with low influences: invariance and optimality. Annals of Mathematics, 171(1):295-341.
Mossel, Elchanan, Ryan, O'Donnell, Oded, Regev, Jeffrey, Steif, and Benjamin, Sudakov (2006) Non-interactive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami-Beckner inequality. Israel Journal of Mathematics, 154:299-336.
Mossel, Elchanan, Ryan, O'Donnell, and Rocco, Servedio (2004) Learning functions of krelevant variables. Journal of Computer and System Sciences, 69(3):421-434.
Mossel, Elchanan, Krzysztof, Oleszkiewicz, and Arnab, Sen (2012) On reverse hyper-contractivity. Tech. Rep. 1108.1210, arXiv.
Muller, David (1954a) Application of Boolean algebra to switching circuit design and to error detection. IRE Transactions on Electronic Computers, 3(6):6-12.
Muller, David (1954b) Boolean algebras in electric circuit design. American Mathematical Monthly, 61(7):27-28.
Müller, Paul (2005) Isomorphisms between H1 spaces, vol. 66 of Monografie Matematyczne. Birkhauser.
Nakashima, Akira (September 1935) The theory of relay circuit composition. Journal of the Institute of Telegraph and Telephone Engineers of Japan, 150:731-752.
Naor, Joseph, and Moni, Naor (1993) Small-bias probability spaces: efficient constructions and applications. SIAM Journal on Computing, 22(4):838-856.
Nazarov, Fedor (2003) On the maximal perimeter of a convex set in Rn with respect to a Gaussian measure. In: Geometric Aspects of Functional Analysis, Israel Seminar, vol. 1807. pp. 169-187.
Nazarov, Fedor, and Anatoliy, Podkorytov (2000) Ball Haagerup, and distribution functions. Complex Analysis, Operators, and Related Topics. Operator Theory: Advances and Applications, 113:247-267.
Nazarov, Fedor, Mikhail, Sodin, and Alexander, Vol'berg (2002) The geometric Kannan-Lovász-Simonovits lemma, dimension-free estimates for volumes of sublevel sets of polynomials, and distribution of zeros of random analytic functions. Algebra i Analiz, 14(2):214-234.
Nelson, Edward (1966) A quartic interaction in two dimensions. In: Mathematical Theory of Elementary Particles, MIT Press. pp. 69-73.
Nelson, Edward (1973) The free Markoff field. Journal of Functional Analysis, 12:211-227.
Neveu, Jacques (1976) Sur l'espérance conditionnelle par rapport à un mouvement brownien. Annales de l'Institut Henri Poincaré (B), 12(2):105-109.
Ninomiya, Ichizo (1958) A theory of the coordinate representations of switching functions. Memoirs of the Faculty of Engineering, Nagoya University, 10:175-190.
Nisan, Noam, and Mario, Szegedy (1994) On the degree of Boolean functions as real polynomials. Computational Complexity, 4(4):301-313.
Nisan, Noam, and Avi, Wigderson (1995) On rank vs. communication complexity. Combinatorica, 15(4):557-565.
O'Donnell, Ryan (2003) Computational applications of noise sensitivity. Ph.D. thesis, Massachusetts Institute of Technology.
O'Donnell, Ryan (2004) Hardness amplification within NP. Journal of Computer and System Sciences, 69(1):68-94.
O'Donnell, Ryan, Michael, Saks, Oded, Schramm, and Rocco, Servedio (2005) Every decision tree has an influential variable. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science. pp. 31-39.
O'Donnell, Ryan, and Rocco, Servedio (2006) Learning monotone decision trees in polynomial time. In: Proceedings of the 21st Annual IEEE Conference on Computational Complexity. pp. 213-225.
O'Donnell, Ryan, and Rocco, Servedio (2007) Learning monotone decision trees in polynomial time. SIAM Journal on Computing, 37(3):827-844.
O'Donnell, Ryan, and Rocco, Servedio (2008) Extremal properties of polynomial threshold functions. Journal of Computer and System Sciences, 74(3):298-312.
O'Donnell, Ryan, and Karl, Wimmer (2013) Sharpness of KKL on Schreier graphs. Electronic Communications in Probability, 18:1-12.
O'Donnell, Ryan, and John, Wright (2012) A new point of NP-hardness for Unique-Games. In: Proceedings of the 44th Annual ACM Symposium on Theory of Computing. pp. 289-306.
O'Donnell, Ryan, and Yi, Wu (2008) An optimal SDP algorithm for Max-Cut, and equally optimal Long Code tests. In: Proceedings of the 40th Annual ACMSymposium on Theory of Computing. pp. 335-344.
O'Donnell, Ryan, and Yi, Wu (2009) 3-bit Dictator testing: 1 vs. 5/8. In: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms. pp. 365-373.
Oleszkiewicz, Krzysztof (2003) On a nonsymmetric version of the Khinchine-Kahane inequality. In: Stochastic inequalities and applications, ed. Evariste, Giné, Christian, Houdré, and David, Nualart, Birkhäuser, vol. 56. pp. 157-168.
Paley, Raymond (1932) A remarkable series of orthogonal functions (I). Proceedings of the London Mathematical Society, 2(1):241-264.
Parnas, Michael, Dana, Ron, and Alex, Samorodnitsky (2001) Proclaiming dictators and juntas or testing Boolean formulae. In: Proceedings of the 5th Annual International Workshop on Randomized Techniques in Computation. pp. 273-284.
Parnas, Michal, Dana, Ron, and Alex, Samorodnitsky (2002) Testing basic Boolean formulae. SIAM Journal on Discrete Mathematics, 16(1):20-46.
Penrose, Lionel (1946) The elementary statistics of majority voting. Journal of the Royal Statistical Society, 109(1):53-57.
Peralta, René (1990) On the randomness complexity of algorithms. Tech. Rep. 90-1, University of Wisconsin, Milwaukee.
Peres, Yuval (2004) Noise stability of weighted majority. Math/0412377, arXiv.
Pisier, Gilles (1986) Probabilistic methods in the geometry of Banach spaces. In: Probability and analysis (Varenna, 1985), Springer, Berlin, vol. 1206 of Lecture Notes in Mathematics. pp. 167-241.
Pisier, Gilles, and Joel, Zinn (1978) On the limit theorems for random variables with values in the spaces Lp(2 ≤ p < ∞). Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 41(4):289-304.
Raghavendra, Prasad (2008) Optimal algorithms and inapproximability results for every CSP? In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing. pp. 245-254.
Raghavendra, Prasad (2009) Approximating NP-hardproblems: efficient algorithms and their limits. Ph.D. thesis, University of Washington.
Rao, Calyampudi (1947) Factorial experiments derivable from combinatorial arrangements of arrays. Journal of the Royal Statistical Society, 9(1):128-139.
Razborov, Alexander (1993) Bounded arithmetic and lower bounds in Boolean complexity. In: Feasible Mathematics II, ed. Peter, Clote and Jeffrey, Remmel, Birkhäuser. pp. 344-386.
Riker, William (1961) Voting and the summation of preferences: an interpretive bibliographic review of selected developments during the last decade. American Political Science Review, 55(4):900-911.
Rinott, Yosef, and Vladimir, Rotar' (2001) A remark on quadrant normal probabilities in high dimensions. Statistics & Probability Letters, 51(1):47-51.
Rosenthal, Haskell (1976) Convolution by a biased coin. In: The Altgeld Book 1975/1976, University of Illinois, pp. II.1-II.17.
Rossignol, Raphaël (2006) Threshold for monotone symmetric properties through a logarithmic Sobolev inequality. Annals of Probability, 34(5):1707-1725.
Rotar', Vladimir (1973) Some limit theorems for polynomials of second order. Teoriya Veroyatnostei i ee Primeneniya, 18(3):527-534.
Rotar', Vladimir (1974) Some limit theorems for polynomials of second degree. Theory of Probability and Its Applications, 18(3):499-507.
Rotar', Vladimir (1975) Limit theorems for multilinear forms and quasipolynomial functions. Teoriya Veroyatnostei i ee Primeneniya, 20(3):527-546.
Rotar', Vladimir (1979) Limit theorems for polylinear forms. Journal of Multivariate Analysis, 9(4):511-530.
Roth, Alvin, ed. (1988) The Shapley Value: Essays in Honor ofLloyd S. Shapley. Cambridge University Press.
Roth, Klaus (1953) On certain sets of integers. Journal of the London Mathematical Society, 28(1):104-109.
Rothaus, Oscar (1976) On “bent” functions. Journal of Combinatorial Theory, Series A, 20(3):300-305.
Rousseau, Jean-Jacques (1762) Du contrat social. Marc-Michel Rey.
Rubin, Herman, and Richard, Vitale (1980) Asymptotic distribution of symmetric statistics. Annals of Statistics, 8(1):165-170.
Rubinfeld, Ronitt, and Madhu, Sudan (1996) Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing, 25(2):252-271.
Rudin, Walter (1962) Fourier Analysis on Groups. John Wiley & Sons.
Russo, Lucio (1981) On the critical percolation probabilities. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 56(2):229-237.
Russo, Lucio (1982) An approximate zero-one law. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 61(1):129-139.
Saeki, Sadahiro (1968) On norms of idempotent measures. Proceedings of the American Mathematical Society, 19(3):600-602.
Saks, Michael, and Avi, Wigderson (1986) Probabilistic Boolean decision trees and the complexity of evaluating game trees. In: Proceedings of the 27th Annual IEEE Symposium on Foundations of Computer Science. pp. 29-38.
Schmidt, Erhard (1948) Die Brunn-Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und nichteuklidischen Geometrie. I. Mathematische Nachrichten, 1:81-157.
Schramm, Oded, and Jeffrey, Steif (2010) Quantitative noise sensitivity and exceptional times for percolation. Annals of Mathematics, 171(2):619-672.
Schreiber, Michel (1967) Fermeture en probabilité des chaos de Wiener. Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, Séries A, 265:859-861.
Schreiber, Michel (1969) Fermeture en probabilité de certains sous-espaces d'un espace L2. Application aux chaos de Wiener. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 14:36-48.
Segal, Irving (1970) Construction of non-linear local quantum processes: I. Annals of Mathematics, 92:462-481.
Shannon, Claude (1937) A Symbolic Analysis ofRelay and Switching Circuits. Master's thesis, Massachusetts Institute of Technology.
Shapley, Lloyd (1953) A value for n-person games. In: Contributions in the Theory of Games, volume II, ed. Harold, Kuhn and Albert, Tucker, Princeton University Press. pp. 307-317.
Sheppard, William (1899) On the application of the theory of error to cases of normal distribution and normal correlation. Philosophical Transactions of the Royal Society of London, Series A, 192:101-167, 531.
Sherman, Jonah (2008) The randomized decision tree complexity of the recursive majority of three function on 3n inputs is at least 2.5n. Unpublished.
Shestakov, Victor (1938) Some Mathematical Methods for the Construction and Simplification of Two-Terminal Electrical Networks of Class A. Ph.D. thesis, Lomonosov State University.
Shevtsova, Irina (2013) On the absolute constants in the Berry–Esseen inequality and its structural and nonuniform improvements. Informatika i Ee Primeneniya, 7(1):124-125.
Siegenthaler, Thomas (1984) Correlation-immunity of nonlinear combining functions for cryptographic applications. IEEE Transactions on Information Theory, 30(5):776-780.
Simon, Barry, and Raphael, Høegh-Krohn (1972) Hypercontractive semigroups and two dimensional self-coupled Bose fields. Journal of Functional Analysis, 9:121-180.
Siu, Kai-Yeung, and Jehoshua, Bruck (1991) On the power of threshold circuits with small weights. SIAM Journal on Discrete Mathematics, 4(3):423-435.
Steele, J. Michael (1986) An Efron-Stein inequality for nonsymmetric statistics. Annals of Statistics, 14(2):753-758.
Stein, Charles (1972) A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In: Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability. University of California Press. pp. 583-602.
Stein, Charles (1986) Approximate computation ofexpectations. Institute of Mathemat-ical Statistics Lecture Notes. Institute of Mathematical Statistics, Hayward, CA.
Stothers, Andrew (2010) On the complexity ofmatrix multiplication. Ph.D. thesis, University of Edinburgh.
Subbotovskaya, Bella (1961) Realizations of linear functions by formulas using V,&, -. Doklady Akademii Nauk SSSR, 136(3):553-555.
Sudakov, Vladimir, and Boris, Tsirel'son (1978) Extremal properties of half-spaces for spherically invariant measures. Journal of Soviet Mathematics, 9(1):9-18.
Originally published in Zap. Nauchn. Sem. Leningrad. Otdel. Math. Inst. Steklova., 41:14-21, 1974.
Szulga, Jerzy (1998) Introduction to Random Chaos. Chapman & Hall.
Takemura, Akimichi (1983) Tensor Analysis of ANOVA Decomposition. Journal of the American Statistical Association, 78(384):894-900.
Talagrand, Michel (1989) A conjecture on convolution operators, and a non-Dunford-Pettis operator on L1. Israel Journal of Mathematics, 68(1):82-88.
Talagrand, Michel (1993) Isoperimetry, logarithmic Sobolev inequalities on the discrete cube and Margulis' graph connectivity theorem. Geometric and Functional Analysis, 3(3):298-314.
Talagrand, Michel (1994) On Russo's approximate zero-one law. Annals of Probability, 22(3):1576-1587.
Talagrand, Michel (1996) How much are increasing sets positively correlated?Combinatorica, 16(2):243-258.
Talagrand, Michel (2006) Regularization from L1 by convolution. http://www.math.jussieu.fr/talagran/prizes/convolution.pdf.
Tannenbaum, Meyer (1961) The establishment ofa unique representation for a linearly separable function. Tech. rep., Lockheed Missiles and Space Company. Threshold Switching Techniques, 20:1-5.
Tardos, Gabor (1989) Query complexity, or why is it difficult to separate NPA ∩ coNPA from PA by random oracles?Combinatorica, 9(4):385-392.
Terras, Audrey (1999) Fourier Analysis on Finite Groups and Applications. Cambridge University Press.
Teuwen, Jonas (2012) A cornucopia of Hermite polynomials. http://fa.its.tudelft.nl/~teuwen/Writings/Proof-of-competency.pdf.
Thomason, Andrew (1987) Pseudo-random graphs. Annals of Discrete Mathematics, 144:307-331.
Tillich, Jean-Pierre, and Gilles, Zemor (2000) Discrete isoperimetric inequalities and the probability of a decoding error. Combinatorics, Probability and Computing, 9(5):465-479.
Titsworth, Robert (1962) Correlation properties ofcyclic sequences. Ph.D. thesis, California Institute of Technology.
Titsworth, Robert (1963) Optimal ranging codes. Tech. Rep. 32-411, Jet Propulsion Laboratory.
Trevisan, Luca, Gregory, Sorkin, Madhu, Sudan, and David, Williamson (2000): Gadgets, approximation, and linear programming. SIAM Journal on Computing, 29(6):2074-2097.
Trotter, Hale (1958) Approximation of semi-groups of operators. Pacific Journal of Mathematics, 8:887-919.
Uhlenbeck, George, and Leonard, Ornstein (1930) On the theory of the Brownian motion. Physical Review, 36(5):823-841.
Valiant, Gregory (2012) Finding correlations in subquadratic time, with applications to learning parities and juntas with noise. Tech. Rep. TR12-006, Electronic Colloquium on Computational Complexity.
Valiant, Leslie (1984) A theory of the learnable. Communications of the ACM, 27(11):1134-1142.
Vassilevska Williams, Virginia (2012) Multiplying matrices faster than Coppersmith-Winograd. In: Proceedings of the 44th Annual ACM Symposium on Theory of Computing. pp. 887-898.
Vilenkin, Naum (1947) On a class of complete orthonormal systems. Izvestiya Rossiiskoi Akademii Nauk, Seriya Matematicheskaya, 11(4):363-400.
Viola, Emanuele (2009) Correlation bounds for polynomials over {0, 1}. SIGACT News, 40(1):27-44.
Vitale, Richard (1984) An expansion for symmetric statistics and the Efron-Stein inequality. In: Inequalities in Statistics and Probability, Institute of Mathematical Statistics, vol. 5 of Lecture Notes—Monograph Series. pp. 112-114.
von Mises, Richard (1947) On the asymptotic distribution of differentiable statistical functions. Annals of Mathematical Statistics, 18(3):309-348.
Walsh, Joseph (1923) A closed set of normal orthogonal functions. American Journal of Mathematics, 45(1):5-24.
Weissler, Fred (1979) Two-point inequalities, the Hermite semigroup, and the Gauss-Weierstrass semigroup. Journal of Functional Analysis, 32(1):102-121.
Weissler, Fred (1980) Logarithmic Sobolev inequalities and hypercontractive estimates on the circle. Journal of Functional Analysis, 37(2):218-234.
Wolff, Pawel (2007) Hypercontractivity of simple random variables. Studia Mathematica, 180(3):219-236.
Xiao, Guozhen, and James, Massey (1988) A spectral characterization of correlation-immune combining functions. IEEE Transactions on Information Theory, 34(3):569-571.
Yang, Ke (2004) On the (im)possibility of non-interactive correlation distillation. In: Proceedings of the 6th Annual Latin American Informatics Symposium. pp. 222-231.
Yao, Andrew (1977) Probabilistic computations: Towards a unified measure of complexity. In: Proceedings of the 9th Annual ACM Symposium on Theory of Computing. pp. 222-227.
Yao, Andrew (1985) Separating the polynomial time hierarchy by oracles. In: Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science. pp. 1-10.
Zhegalkin, Ivan (1927) On a technique of calculating propositions in symbolic logic. Matematicheskii Sbornik, 43:9-28.
Zuev, Yuri (1989) Asymptotics of the logarithm of the number of threshold functions of the algebra of logic. Doklady Akademii Nauk SSSR, 39(3):512-513.
Zwick, Uri (1999) Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems. In: Proceedings of the 31st Annual ACM Symposium on Theory of Computing. pp. 679-687.
Zwick, Uri (2002) Computer assisted proof of optimal approximability results. In: Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms. pp. 496-505.

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