The properties of general Fourier partial sums of functions $f \in C_L$

TL;DR

The paper addresses when Fourier partial sums with respect to general orthonormal systems are bounded or converge for functions in the Lip1 (and thus $C_L$) class. It develops a framework using Fourier coefficients $C_n(f)$, partial sums $S_n$, and auxiliary quantities $M_n(x)$, $B_n$, and $Q_n$, to relate ONS properties to pointwise behavior on $[0,1]$. The main results, Theorems $t2$ and $t3$, establish boundedness under specific $p,q$ in $E(oldsymbol{ obreak\varphi})$ (with $q=1$, $p=u$) and prove sharpness by constructing Lip1 functions with divergence when $M_n(t)$ is unbounded on a set $G$ of full measure; Banach-Steinhaus arguments yield further Lip1 functions with divergent partial sums. The paper also analyzes efficiency for standard systems (e.g., trigonometric and Haar) where $M_n(x)=O(1)$, and shows that every ONS contains a subsystem with almost everywhere convergence for all $f ext{ in }Lip1$, providing precise criteria for convergence behavior in generalized Fourier series.

Abstract

In this paper, we investigated the Fourier partial sums with respect to general orthonormal systems when the function $f$ belongs to some differentiable class of functions

Theorems & Definitions (20)

• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Definition 1