Walsh-Fourier coefficients and locally constant functions

Wade, William R.

doi:10.1090/S0002-9939-1983-0684633-3

Walsh-Fourier coefficients and locally constant functions
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Proc. Amer. Math. Soc. 87 (1983), 434-438 Request permission

Abstract:

A condition on the Walsh-Fourier coefficients of a continuous function $f$ sufficient to conclude that $f$ is locally constant is obtained. The condition contains certain conditions identified earlier by Bočkarev, Coury, Skvorcov and Wade, and Powell and Wade.

References

On the Fourier-Walsh coefficients

4

J. E. Coury, Walsh series with coefficients tending monotonically to zero, Pacific J. Math. 54 (1974), no. 2, 1–16. MR 372520
N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372–414. MR 32833, DOI 10.1090/S0002-9947-1949-0032833-2
Charles H. Powell and William R. Wade, Term by term dyadic differentiation, Canadian J. Math. 33 (1981), no. 1, 247–256. MR 608868, DOI 10.4153/CJM-1981-020-2

Theory of the integral

V. A. Skvorcov and W. R. Wade, Generalization of some results concerning Walsh series and the dyadic derivative, Anal. Math. 5 (1979), no. 3, 249–255 (English, with Russian summary). MR 549241, DOI 10.1007/BF01908907
J. L. Walsh, A Closed Set of Normal Orthogonal Functions, Amer. J. Math. 45 (1923), no. 1, 5–24. MR 1506485, DOI 10.2307/2387224