Decay of Walsh series and dyadic differentiation

Author:
William R. Wade

Journal:
Trans. Amer. Math. Soc. **277** (1983), 413-420

MSC:
Primary 42C10; Secondary 43A75

MathSciNet review:
690060

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the partial sums of the Walsh-Fourier series of an integrable function . Let represent the ratio , for , and let represent the function . We prove that belongs to for all . We observe, using inequalities of Paley and Sunouchi, that the operator arises naturally in connection with dyadic differentiation. Namely, if is strongly dyadically differentiable (with derivative ) and has average zero on the interval [0, 1], then the norms of and are equivalent when . We improve inequalities implicit in Sunouchi's work for the case and indicate how they can be used to estimate the norm of and the dyadic norm of by means of mixed norms of certain random Walsh series. An application of these estimates establishes that if is strongly dyadically differentiable in dyadic , then .

**[1]**P. L. Butzer and H. J. Wagner,*Walsh-Fourier series and the concept of a derivative*, Applicable Anal.**3**(1973), 29–46. Collection of articles dedicated to Eberhard Hopf on the occasion of his 70th birthday. MR**0404978****[2]**P. L. Butzer and H. J. Wagner,*On dyadic analysis based on the pointwise dyadic derivative*, Anal. Math.**1**(1975), no. 3, 171–196 (English, with Russian summary). MR**0404979****[3]**N. J. Fine,*On the Walsh functions*, Trans. Amer. Math. Soc.**65**(1949), 372–414. MR**0032833**, 10.1090/S0002-9947-1949-0032833-2**[4]**Adriano M. Garsia,*Martingale inequalities: Seminar notes on recent progress*, W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1973. Mathematics Lecture Notes Series. MR**0448538****[5]**N. R. Ladhawala,*Absolute summability of Walsh-Fourier series*, Pacific J. Math.**65**(1976), no. 1, 103–108. MR**0417678****[6]**J. Marcinkiewicz,*Sur les multiplicateurs des séries de Fourier*, Studia Math.**8**(1939), 79-91.**[7]**R. E. A. C. Paley,*A remarcable series of orthogonal functions*. I, Proc. London Math. Soc.**34**(1931), 241-264.**[8]**Gen-Ichirô Sunouchi,*On the Walsh-Kaczmarz series*, Proc. Amer. Math. Soc.**2**(1951), 5–11. MR**0041259**, 10.1090/S0002-9939-1951-0041259-1**[9]**A. Zygmund,*On the convergence and summability of power series in the circle of convergence*. I, Fund. Math.**30**(1938), 170-196.**[10]**-,*Trigonometric series*, Vol. I, Cambridge Univ. Press, Cambridge, 1959.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
42C10,
43A75

Retrieve articles in all journals with MSC: 42C10, 43A75

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1983-0690060-X

Keywords:
Walsh series,
dyadic derivative,
square function,
maximal function,
dyadic

Article copyright:
© Copyright 1983
American Mathematical Society