Chebyshev systems and zeros of a function on a convex curve

TL;DR

This paper generalizes classical zero and sign change theorems for functions orthogonal to Chebyshev systems on convex curves, extending results like Hurwitz's theorem and the four-vertex theorem to higher dimensions and discrete settings.

Contribution

It introduces a generalized zero theorem for functions orthogonal to Chebyshev systems on convex curves in Euclidean space, extending classical results to higher dimensions and discrete analogs.

Findings

Zero bounds are sharp and attained for classical Chebyshev systems.

Theorems extend Hurwitz's and four-vertex theorems to convex curves in multiple dimensions.

Discrete analogs provide versions for convex polygonal lines.

Abstract

The classical Hurwitz theorem says that if n first "harmonics" (2n + 1 Fourier coefficients) of a continuous function f(x) on the unit circle are zero, then f(x) changes sign at least 2n + 1 times. We show that similar facts and its converse hold for any function that are orthogonal to a Chebyshev system. These theorems can be extended for convex curves in d-dimensional Euclidean space. Namely, if a function on a curve is orthogonal to the space of n-degree polynomials, then the function has at least nd + 1 zeros. This bound is sharp and is attained for curves on which the space of polynomials forms classical polynomial and trigonometric Chebyshev systems. We can regard the theorem of zeros as a generalization of the four-vertex theorem. There exists a discrete analog of the theorem of zeros for convex polygonal lines which yields a discrete version of the four-vertex theorem.