A REMARK ON THE ARCSINE DISTRIBUTION AND THE HILBERT TRANSFORM

Area of Science:

• Mathematical analysis
• Real and complex analysis
• Approximation theory

Background:

• Orthogonal polynomials are fundamental in mathematical analysis.
• The roots of orthogonal polynomials often exhibit specific distributions, such as the arcsine distribution for weights on [-1,1].
• The Hilbert transform is a key operator in harmonic analysis with significant applications.

Purpose of the Study:

• To establish a connection between the distribution of orthogonal polynomial roots and the properties of the Hilbert transform.
• To investigate conditions under which a function, related to the arcsine distribution, possesses a vanishing Hilbert transform.
• To introduce and prove a novel localized Parseval-type identity.

Main Methods:

• Analysis of orthogonal polynomials in L^2 spaces with specific weight functions.
• Application of Tricomi's theorem concerning the Hilbert transform.
• Derivation of a localized Parseval-type identity for functions in L^2(-1,1).

Main Results:

• Demonstrated that if a function f(x)(1-x^2)^1 is in L^2(-1,1) and its Hilbert transform vanishes on (-1,1), then f(x) is a multiple of the arcsine distribution.
• Proved a new localized Parseval-type identity for functions f(x)(1-x^2)^1 in L^2(-1,1) with a mean value of 0.

Conclusions:

• The study provides a deeper understanding of the relationship between orthogonal polynomials, root distributions, and Hilbert transforms.
• The findings extend existing results on Hilbert transforms and introduce new identities with potential applications in mathematical physics and signal processing.