Abstract
It is well known that the Lp norm is an essential tool in the study of Fourier series. In letting p + co the Lp norm becomes the essential upper bound and Lp behavior formally becomes Lipschitz behavior. It is a natural way to generalize results on Fourier series by replacing the powerfunction by more general classes of functions. In this paper we shall focus on the application of a class of functions which has been used extensively in classical analysis by a number of authors [S, 9, 12, 133. Following the notation in [6] we denote this class by Y[a, b] (a < b), which is defined as the collection of all positive functions @ defined on (0, co) such that @(u)/u” is nondecreasing and @(u)/u” is nonincreasing. The class Y[a, b] is known to be equivalent to other class of functions of the “power type” such as those introduced by Marcinkiewicz [ 111, Koizumi [lo] , Woyczynski [ 151, and the class of R 0 varying functions [ 141. For the details the readers are referred to [4, 6, 71. The author [4] has proved the following integral relationship for functions in Y[a, h]:
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