On conjugate functions and trigonometric series

Abstract

The present paper is an abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Physical and Mathematical Sciences.

On the Localization of Rectangular Partial Sums for Multiple Fourier Series

“…a functions on Tn over subintervals of Tn. As a consequence we obtain the uniform convergence for the rectangular partial sums of functions in W\, p>n. It has been shown in [7] that the DiniLipschitz theorem holds on T2 for the rectangular partial sum. Since the method of the proof in [7] is unnecessarily complicated and it is not clear that the method employed there can be applied to the higher dimensional cases, we will indicate at the appropriate place that the corresponding theorem in T" actually follows along the lines of the arguments in §2.…”

On some properties of multiple conjugate trigonometric series

“…The estimate (7) is more precise than the estimate (2.1.14) in [1] and it can be proved by arguments analogous to those in [2], pp.157-160.…”

“…n ), n ∈ N, n > 1, be a function, 2π-periodic in each variable, σ n [f ] its n-multiple trigonometric Fourier series, andσ n [f ] its conjugate series with respect to n variables (see, e.g., [1]). …”

“…There is another lemma in [1] (see p.171, Lemma 11), which can be corrected by means of (7) as follows:…”

“…. , x n )-type were for the first time introduced and applied in the works of L.Zhizhiashvili (see [1], [5]). (x 1 , . .…”