Abstract
Let S j : (Ω, P) → S 1 ⊂ ℂ be an i.i.d. sequence of Steinhaus random variables, i.e. variables which are uniformly distributed on the circle S 1. We determine the best constants a p in the Khintchine-type inequality
for 0 < p < 1, verifying a conjecture of U. Haagerup that
. Both expressions are equal for p = p 0 }~ 0.4756. For p ≥ 1 the best constants a p have been known for some time. The result implies for a norm 1 sequence x ∈ ℂn, ‖x‖2 = 1, that
, answering a question of A. Baernstein and R. Culverhouse.
Similar content being viewed by others
References
E. Artin, The Gamma function, Holt, Rinehart and Winston, London, 1964.
A. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover Publ., New York, 1972.
A. Baernstein and R. C. Culverhouse, Majorization of sequences, sharp vector Khintchine inequalities, and bisubharmonic functions, Studia Mathematica 152 (2002), 221–248.
T. Figiel, P. Hitczenko, W. B. Johnson, G. Schechtmen and J. Zinn, Extremal properties of Rademacher functions with applications to the Khintchine and Rosenthal inequalities, Transactions of the American Mathematical Society 349 (1977), 997–1027.
I. Gradshtein and I. Ryzhik, Table of Integrals, Series and Products, 5th edition, Academic Press, New York, 1984.
U. Haagerup, The best constants in the Khintchine inequality, Studia Mathematica 70 (1982), 231–283.
H. König and S. Kwapień, Best Khintchine type inequalities for sums of independent, rotationally invariant random vectors, Positivity 5 (2001), 115–152.
W. Magnus and F. Oberhettinger, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd edition, Springer, Berlin, 1966.
F.L. Nazarov and A.N. Podkorytov, Ball, Haagerup, and distribution functions, in Complex Analysis, Operators and Related Topics, (V. Havin and N. Nikolski, eds.), Birkhäuser, Basel, 2000, pp. 247–267.
A. Pełczyński, Norms of classical operators in function spaces, Colloq. L. Schwartz, Astérisque 131 (1985), 107–126.
J. Sawa, On the best constant in the Khintchine inequality for complex Steinhaus variables, the case p = 1, Studia Mathematica 81 (1985), 107–126.
S. Szarek, On the best constants in the Khintchine inequality, Studia Mathematica 58 (1976), 197–208.
G. N. Watson, A Treatise on the Theory of Bessel Functions, 5th edition, Cambridge University Press, 1952.
Author information
Authors and Affiliations
Corresponding author
Additional information
Gratefully dedicated to Joram Lindenstrauss whose fundamental work and professional integrity inspired a generation of mathematicians
Rights and permissions
About this article
Cite this article
König, H. On the best constants in the Khintchine inequality for Steinhaus variables. Isr. J. Math. 203, 23–57 (2014). https://doi.org/10.1007/s11856-013-0006-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-013-0006-y