Matrix transformations of sequences and applications in Fourier analysis

Area of Science:

• Mathematical Analysis
• Operator Theory
• Sequence Theory

Background:

• Nörlund means are a fundamental concept in sequence theory.
• Matrix transformations provide a method for generalizing sequence means.
• Fourier series theory has well-established theorems regarding convergence.

Purpose of the Study:

• To define and analyze generalized matrix means derived from Nörlund means.
• To establish sufficient conditions for the convergence of these generalized means.
• To extend the applicability of existing theorems from Fourier series theory.

Main Methods:

• Consideration of a sequence and its associated Nörlund mean.
• Derivation of a generalized mean using matrix transformations.
• Analysis of sufficient conditions for the convergence of the generalized matrix means.

Main Results:

• Sufficient conditions for the convergence of generalized matrix means from Nörlund ones were identified.
• The study establishes a link between Nörlund means and generalized matrix means.
• The findings provide a theoretical framework for transferring theorems.

Conclusions:

• The derived conditions facilitate the convergence analysis of generalized matrix means.
• The research enables the extension of classical Fourier series theorems to more general operator sequences.
• This work contributes to a deeper understanding of convergence properties in generalized sequence spaces.