William R. Wade

Cesàro summability of double Walsh-Fourier series

We introduce quasi-local operators (these include operators of Calder6n-Zygmund type), a hybrid Hardy space HO of functions of two variables, and we obtain sufficient conditions for a quasi-local maximal operator to be of weak type ( , 1) . As an application, we show that Cesatro means of the double Walsh-Fourier series of a function f converge a.e. when f belongs to HO. We also obtain the dyadic analogue of a summability result of Marcienkiewicz and Zygmund valid for all f E L1 provided summability takes place in some positive cone.

Recent developments in the theory of Walsh series

We survey research done on the theory of Walsh series during the decade 1971-1981. Particular attention is given to convergence of Walsh-Fourier series, gap Walsh series, growth of Walsh-Fourier coefficients, dyadic differentiation, and uniqueness of Walsh series.

Decay of Walsh series and dyadic differentiation

Let W2n [ f ] denote the 2 th partial sums of the Walsh-Fourier series of an integrable functionf. Let pn(x) represent the ratio W2n[ f, x]/2 , for x E [0, 1], and let T(f) represent the function (pn2)1/2. We prove that T(f) belongs to LP[0, 1] for all 0 <p < oo. We observe, using inequalities of Paley and Sunouchi, that the operatorf -T(f ) arises naturally in connection with dyadic differentiation. Namely, if f is strongly dyadically differentiable (with derivative Df ) and has average zero on the interval [0, 1], then the LP norms of f and T(Df) are equivalent when 1 < p < oo. We improve inequalities implicit in Sunouchis work for the case p = 1 and indicate how they can be used to estimate the L1 norm of T(Df ) and the dyadic H1 norm of f by means of mixed norms of certain random Walsh series. An application of these estimates establishes that if f is strongly dyadically differentiable in dyadic H1, then Jj12N= II WN[f, x] GN[f, x]/Ndx C o.

Recursions That Produce Pythagorean Triples

Peter W. Wade, the son of this father-son team, began his study of higher mathematics at age 9 when he discovered how to factor the difference of two squares while memorizing his multiplication tables. After receiving his B.S. and M.S. degrees from the University of Tennessee in 1992 and 1994, he got his first teaching job at Martin Luther King magnet school in Nashville. He is presently employed at Page High School, in Franklin, Tennessee. His outside interests include professional hockey and cinemaphotography (his work appears regularly on CATV Channel 19 in Nashville).

AN ANALOGUE OF A THEOREM OF FERENC LUKÁCS FOR DOUBLE WALSH—FOURIER SERIES

AbstractA theorem of Ferenc Lukács states that if a periodic function

A Walsh system for polar coordinates

f

Uniqueness of Haar series which are

is integrable in Lebesgues sense and has a discontinuity of first kind at some point