On multiplicative Jacobi polynomials and function approximation through multiplicative series

TL;DR

This paper introduces multiplicative Jacobi polynomials derived from a multiplicative Sturm-Liouville equation, extending classical polynomial properties and demonstrating their use in function approximation via multiplicative Fourier series.

Contribution

It develops the theory of multiplicative Jacobi polynomials, extending classical properties, and shows their application in representing and approximating functions through multiplicative Fourier series.

Findings

Multiplicative Jacobi polynomials are solutions to a specific Sturm-Liouville problem.

Classical properties of Jacobi polynomials are extended to the multiplicative case.

Any positive function can be represented and approximated by multiplicative Jacobi-Fourier series.

Abstract

In this contribution, we introduce the multiplicative Jacobi polynomials that arise as one of the solutions of the multiplicative Sturm-Liouville equation \begin{equation*} \frac{d^*}{dx}\left( e^{(1-x^2)\omega(x)}\odot \frac{d^*y}{dx} \right)\oplus \left(e^{ n(n+\alpha+\beta+1)\omega(x)}\odot y\right)=1, \ x\in[-1,1], \end{equation*} where $\omega(x)=(1-x)^{\alpha}(1+x)^{\beta}$ with $\alpha, \beta >-1$ real numbers and $n$ is a non-negative integer number. We extend some properties of classical Jacobi polynomials to the multiplicative case. In particular, we present several properties of multiplicative Legendre polynomials and multiplicative Chebyshev polynomials of first and second kind. We also prove that every real and positive function can be expressed as a multiplicative Jacobi-Fourier series and show that such functions can be approximated by the corresponding partial products of these series. We illustrate the obtained results with some examples.