Cosine polynomials with restrictions on their algebraic representation

TL;DR

This paper demonstrates that for any even algebraic polynomial, one can construct a cosine polynomial with minimal coefficient sum that matches the initial coefficients when expressed as a polynomial in cos x.

Contribution

It establishes a method to approximate even algebraic polynomials with cosine polynomials having restricted algebraic representations.

Findings

Existence of cosine polynomials with small $l_1$-norm matching initial coefficients of any even algebraic polynomial.

Theoretical proof of the approximation capability for algebraic polynomials by cosine polynomials.

Extension of polynomial approximation theory in the context of cosine polynomials with algebraic restrictions.

Abstract

We prove that for any even algebraic polynomial $p$ one can find a cosine polynomial with an arbitrary small $l_1$-norm of coefficients such that the first coefficients of its representation as an algebraic polynomial in $\cos x$ coincide with those of $p$.