Weighted inequalities for generalized polynomials with doubling weights

Area of Science:

• Mathematical Analysis
• Approximation Theory
• Harmonic Analysis

Background:

• Weighted polynomial inequalities like Bernstein and Nikolskii are crucial in approximation theory.
• Previous work by Mastroianni, Totik, and Erdélyi established these for specific polynomial types and weight classes.
• Doubling weights are a significant feature in modern analysis.

Purpose of the Study:

• To extend established weighted polynomial inequalities to the domain of generalized trigonometric polynomials.
• To prove the large sieve inequality for generalized trigonometric polynomials with doubling weights.
• To bridge the gap between classical polynomial inequalities and their trigonometric counterparts.

Main Methods:

• Generalization of classical weighted polynomial inequalities.
• Application of techniques used for proving Bernstein, Marcinkiewicz, Schur, Remez, and Nikolskii inequalities.
• Development of methods for the large sieve inequality in the generalized trigonometric polynomial setting.

Main Results:

• Successful extension of weighted polynomial inequalities to generalized trigonometric polynomials.
• Proof of the large sieve inequality for generalized trigonometric polynomials featuring doubling weights.
• Establishment of new inequalities that expand the scope of existing results.

Conclusions:

• The study successfully generalizes key inequalities, broadening their applicability.
• The results contribute to a deeper understanding of approximation properties of generalized trigonometric polynomials.
• This work opens avenues for further research in related areas of mathematical analysis.