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

@article{citeulike:4168394, 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.}, archivePrefix = {arXiv}, author = {Musin, Oleg R.}, citeulike-article-id = {4168394}, citeulike-linkout-0 = {http://arxiv.org/abs/0903.1908}, citeulike-linkout-1 = {http://arxiv.org/pdf/0903.1908}, eprint = {0903.1908}, keywords = {fewnomials, integral\_geometry, univariate\_functions}, month = {Mar}, posted-at = {2009-03-12 13:34:53}, priority = {3}, title = {Chebyshev systems and zeros of a function on a convex curve}, url = {http://arxiv.org/abs/0903.1908}, year = {2009} }

TY - JOUR ID - citeulike:4168394 L3 - citeulike-article-id:4168394 N2 - 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. TI - Chebyshev systems and zeros of a function on a convex curve KW - fewnomials KW - integral_geometry KW - univariate_functions AU - Musin, Oleg PY - 2009/Mar/11/ UR - http://arxiv.org/abs/0903.1908 ER -