Note on Some Limit Properties of Trigonometric Series of A Certain Type

TL;DR

This paper investigates a specific class of trigonometric series, proves an unverified statement from Zygmund's book, and extends the results to broader contexts, contributing to the understanding of their limit properties.

Contribution

It proves and generalizes an unverified statement about the limit properties of certain trigonometric series from Zygmund's work.

Findings

Proved an unverified statement in Zygmund's book.

Generalized the statement to broader classes of series.

Enhanced understanding of limit behaviors of these series.

Abstract

In this note, we study a certain class of trigonometric series which is important in many problems. An unproved statement in Zygmund's book [5] will be proved and generalized. Further discussions based on this problem will also be made here.