Sufficient conditions on the modulus of continuity ω(δ) are found for an imbedding of function classes Hpω(δ) ⊂ (L). It is shown that in a number of cases these conditions are also necessary.

THE IMBEDDING OF CERTAIN FUNCTION CLASSES Hpω

If / is a function on Ft1 of A-bounded variation and period 2ir, then its nth Fourier coefficient f(n) = 0(1/E" 1/Xj) and its integral modulus of continuity wi(/;<5) = 0(l/s(1/i) 1/X,-). The result on /(n) is best possible in a sense. These results can be extended to certain other classes of functions of generalized variation. For various classes of functions, estimates of the magnitude of the Fourier coeffi- cients and of the integral moduli of continuity can be made. Here we shall do this for certain classes of functions of generalized bounded variation. 1. Let us suppose that / is a real-valued function on an interval 7 of R1. If In = (an, bn) C 7, set f(In) = f(bn) — f(an). We suppose the intervals 7n, n = 1,2,..., to be nonoverlapping. If A = {Xn} is a nondecreasing sequence of positive real numbers such that ^2 1/Xn = oo, we have said (5) that / is of A-bounded variation (ABV) if, for every {7"},

/pdf/on-the-magnitude-of-fourier-coefficients-45vf6he9t4.pdf

On the magnitude of Fourier coefficients

The present paper is a survey of the author’s recent research in the one-dimensional trigonometric series of the type

$$\sum\limits_n {\hat f\left( n \right)} {e^{2\pi i\left( {{n^r}{x_r} + \cdots + n{x_1}} \right)}}.$$

(1.1)

A Class of I.M. Vinogradov’s Series and Its Applications in Harmonic Analysis

In this article we give a brief account of the results on multiple trigonometric and orthogonal series that have been obtained so far and are fundamental from our point of view. We also state results on simple trigonometric and orthogonal series that are new or have been obtained recently.

https://iopscience.iop.org/article/10.1070/RM1973v028n02ABEH001536/pdf

Some Problems in the Theory of Simple and Multiple Trigonometric and Orthogonal Series

In this paper, we prove some new dynamic inequalities on time scales using H¨ older’s inequality and Keller’s chain rule on time scales. These inequalities, as special cases when the time scale T = R and when T = N, contain some generalizations of integral and discrete inequalities due to Hardy, Copson, Leindler and Bennett.

Masako Izumi

Papers

On absolute convergence of Fourier series

On some inequalities for Fourier series

Modulus of continuity of functions defined by trigonometric series

Absolute convergence of Fourier series of convolution functions

On hardy's inequality and its generalization