Maximal theorems for some orthogonal series. I

Abstract

The idea of deducing properties of an orthogonal expansion with respect to one orthonormal sequence from the orthogonal expansion with respect to another orthonormal sequence or of relating these expansions has long been common.For instance, if /~ 2n f(n)un(x) is the orthogonal development, the Marcel Riesz theorem can be reformulated as: let {an} be a sequence of real numbers; then /~ J.n an cos nx for some / e Ap(0, tt), 1 </ < oo, if and only if </> ~ 2n on sin nx for some </> e Ap(0, tt).In other directions there is a large body of equiconvergence or equisummability theorems.Recently, several authors have employed such ideas to obtain important results for expansions with respect to ultraspherical and Jacobi polynomials either by analogy with the corresponding results for Fourier series [14] or by "transplantation" of orthonormal sequences (cf.[2], [4]).In this paper a more axiomatic treatment is given.When {un}, {vn} are sequences of real-valued functions on (0, tt) not necessarily orthonormal or identical, theorems of maximal type are proved relating the properties of kernels : Kk(x, /) = 2 rnke'e-'nt, Jfk(x, /) = 2 rnkun(x)vn(t).n nThe coefficients {rnk} could arise from a summability method, for instance.Under five simple conditions on {un}, {vn} the properties of f * f(t)Kk(x, t)dt, fe A"( -tt, w)0, J -s i.e. for ordinary Fourier series, can be carried over to { 4>(t)^k(x, t) dt, </> e Lp(0, tt), provided /#l,oo ( 1).Although the case p=l is more difficult (perhaps not surprisingly), a slight strengthening of one of the five conditions is all that is required.If now {"}, {vn} are complete and orthonormal, properties of the orthogonal expansions with respect to {un}, {vn} can be read off from the corresponding results for Fourier series ( 2).Typical results are: (i) Let {an} be a sequence of real numbers.Thenf~ 2 an cos nxfor somefe L"(0, tt) if and only ij'<A~2" onun(x)for some t/> e L"(0, tt), pj= 1, oo.