Contributions to the study of Fourier series of function of class Vp

Abstract

In the present thesis we study the different problems concerning the Fourier series of function of Wiener's class Vp . In order to state precisely the results proved, it is necessary to give some definitions and notations. In chapter I, we define the classes Vp and establish its main properties. In section one of chapter II we first study the problem of summability of the sequences, and allied sequences in the classes (Vp1 < p < °°) by infinite matrices which are not necessarily regular. This enables us to obtain the generalizations of some theorems of J. A. Siddiqi [19] [20] which include as a special cases those obtained by Wiener [27], Lozinskii [14], Matveyev cf. [2] and Keogh and Petersen [13]. We also show that under our hypothesis not only above sequences are summable A (or FA) to zero but even the modulii of these sequences are summable A (or FA) to zero. In section 2 of chapter II we study the problem of summability of the sequence {cke ikx} by a normal almost periodic matrix in the class Vp . Thus we find the class of matrices which sums every sequence {Ck} of Fourier-Stieltjes coefficients in the classes Vp(1 < p < °°). It is natural to ask how Fourier coefficients of function of the classes V behave. In chapter III we determine P the order of Fourier coefficients of function of V . We have been P able to find out the best constant which turns out to be V (f) P in our case. We also study how many Fourier coefficients can have exactly the order n . With the help of these estimates we further study the problem of density of positive and negative Fourier sine and cosine coefficients of function of p-th variation. This enables us to obtain the generalization and sharpened version of theorems due to M. and S. Izumi [9]. We generalize their results by replacing (C,I) summability by any method of summability in the strictly larger class (Vp1 < p < °°). Among other things we also show that the hypothesis (10) in case III of theorem D (see page 49 chapter III of this thesis) is superflous and all the results are true uniformity in r = 0, I, 2, ... The chapter IV of this thesis is devoted to the study of certain questions concerning the convergence and absolute convergence of Fourier series of function of the class Vp . Wiener [27] has shown that a Fourier series of a function of Vp (1 < p < °°) converges almost everywhere. In particular he [27] has shown that a Fourier series of a function of the class converges to è [f(x+0) + f(x-O)] for every value of x. Generalizing the above results, we show that a Fourier series of a function of the class Vp (1 < p < °°) converges to 1/2[f(x+0) + f(x-O)] at every point X. Bernstein [3] was one of the first mathematicians who investigated the problem of absolute convergence of Fourier series relating it to the continuity properties of generating fonction in the whole interval of periodicity. This theorem was improved upon in varions ways by authors like Zygmund [33], Szàsz [26]. In section 2 of chapter IV we extend the results of Zygmund and Szàsz on absolute convergence of Fourier series for the fonction of the classes V . P Finally in chapter V, which is the last chapter of this thesis, we discuss the problem of integrability of functions of the class Vp (1 < p < °°). We generalize some integrability theorems of Boas [4] and of Zygmund [33] to functions of the classes Vp (1 < p < °°) We partially answer in affirmative a question posed by Boas [4].