Chebyshev systems and zeros of a function on a convex curve

Details Chebyshev systems and zeros of a function on a convex curve Chebyshev systems and zeros of a function on a convex curve

2009-03-11

arxiv.org/abs/0903.1908v2

Proc. Steklov Inst. of Math., 221 (1998), 236-246

Mathematics

Metric Geometry

Scientific paper

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.

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