Chebyshev systems and zeros of a function on a convex curve

arXiv:0903.1908v2 [math.MG] 27 Apr 2009 Chebyshev systems and zeros of a function on a convex curve Oleg R. Musin 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 simi- lar 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 orthog- onal 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 Cheby- shev 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. 1 Introduction The classical four vertex theorem for an oval (a plane convex closed smooth curve) states that the curvature function on an oval has at least four extrema (vertices) [1]. Recently, a series of works appeared (see [2-6] and others) in which different versions of this theorem are studied for convex curves in Rd (vertices, flat points) and similar phenomena are considered (points of inflection, zeros of higher derivatives etc.). The majority of the results obtained can be interpreted as the lower estimate of the number of zeros of the function which satisfies certain conditions on the curve. In this paper, we tried to define conditions imposed on the function which guarantee the existence of zeros of the function. The L2-orthogonality of a func- tion to a Chebyshev system on an interval or a circle is one of these conditions. (Note that Chebyshev systems of functions have already appeared in Arnold’s paper [2] for similar purposes.) It turns out that if a continuous function on an interval or a circle is L2-orthogonal to a Chebyshev system of functions of order n, then this function has at least n zeros. In this paper, we show that this result is invertible and the lower estimate is attainable for any given Chebyshev system. 1 These results are applicable to functions on convex curves in Rd. A curve in Rd is said to be convex if it intersects any hyperplane at no more than d points while taking their multiplicity in consideration. For plane curves, this definition is consistent with the standard concept of convexity, and the properties of convex curves are now actively studied for d > 2 (see [2, 7, 8] and others). In this paper, a simple connection is revealed between the concept of convexity and Chebyshev systems. Let us consider the restriction of the space of linear functions in Rd to a curve. The curve is convex if and only if this space on the curve is Chebyshev space. The main result of this paper (the theorem of zeros) for convex curves is that 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 estimate is exact 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. We see this by observing that the radius of curvature of a plane convex curve regarded as a function on a circle satisfies the orthogonality condition for n = 1. There exists a discrete analog of the theorem of zeros for convex polygonal lines which yields a discrete version of the four vertex theorem. 2 Chebyshev systems and sign changes Everywhere in this paper we understand a zero of a function as a “stable” zero, i.e., changes of sign of the function. We now provide a more exact definition. The set of zeros of a continuous function on an interval or a circle is a compact set and every connected component of the set of zeros is a point or an interval. We say that a function changes sign on the connected component of the set of zeros if, in any neighborhood of this component, there exist two points at which the values of the function are of different signs. We consider the number of zeros (sign changes) of a function to be the number of connected components of the set of zeros where the function changes sign. If it is an interval, we select one point of each of these connected components and refer to this collection as the zeros (sign changes) of the function. Let us take the union of all connected components of sign changes, and consider the complement. It obviously follows from the definition of a sign change that on every connected component of the complement, the function is of constant sign, i.e., nonnegative or nonpositive, not identically zero. The well-known book by Polya and Szeg¨o [9] contains two statements which are closely connected with this idea. Here are their formulations. Assertion 1 (9, II.4,

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