Abstract
We show that in order for a Walsh series to be locally constant it is necessary for certain blocks of that series to sum to zero. As a consequence, we show that a functionf with a somewhat sparse Walsh—Fourier series is necessarily a Walsh polynomial if its strong dyadic derivative is constant on an interval. In particular, if a Rademacher seriesR is strongly dyadically differentiable and if that derivative is constant on any open subset of [0, 1], thenR is a Rademacher polynomial.
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References
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Wade, W.R. Locally constant dyadic derivatives. Period Math Hung 13, 71–74 (1982). https://doi.org/10.1007/BF01848097
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DOI: https://doi.org/10.1007/BF01848097