Chebyshev systems and zeros of a function on a convex curve
| dc.creator | Musin, Oleg R. | |
| dc.date | 2009-03-11 | |
| dc.date | 2009-04-27 | |
| dc.date.accessioned | 2026-07-07T13:08:20Z | |
| dc.date.available | 2026-07-07T13:08:20Z | |
| dc.description | 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. | |
| dc.identifier | https://arxiv.org/abs/0903.1908 | |
| dc.identifier | http://arxiv.org/abs/0903.1908 | |
| dc.identifier | Proc. Steklov Inst. of Math., 221 (1998), 236-246 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/228388 | |
| dc.subject | Metric Geometry | |
| dc.subject | Classical Analysis and ODEs | |
| dc.title | Chebyshev systems and zeros of a function on a convex curve | |
| dc.type | text |