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Abstract. For a real valued periodic smooth function u on R, n ≥ 0, one defines the osculating polynomial ϕs (of order 2n + 1) at a point s ∈ R to be the unique trigonometric polynomial of degree n, whose value and first 2n derivatives at s coincide with those of u at s. We will say that a point s is a clean maximal flex (resp. clean minimal flex ) of the function u on S 1 if and only if ϕs ≥ u (resp. ϕs ≤ u) and the preimage (ϕ − u) −1 (0) is connected. We prove that any smooth periodic function u has at least n + 1 clean maximal flexes of order 2n + 1 and at least n + 1 clean minimal flexes of order 2n + 1. The assertion is clearly reminiscent of Morse theory and generalizes the classical four vertex theorem for convex plane curves. §1. Introduction For a real valued C 2n -function u on S 1 = R/2πZ, n ≥ 0, one defines the osculating polynomial ϕ s (of order 2n + 1) at a point s ∈ S 1 to be the unique trigonometric polynomial of degree n, ϕ s (t) = a 0 + a 1 cos t + b 1 sin t + · · · + a n cos nt + b n sin nt, whose value and first 2n derivatives at s coincide with those of u at s. If u is C 2n+1 and the value and the first 2n + 1 derivatives of u and ϕ s coincide in s, i.e., if ϕ s hyperosculates u in s, then we call s a flex of u (of order 2n+1). Notice that the order 2n + 1 of the osculating polynomials and flexes in the definition above has been chosen such that it coincides with the dimension of the space A 2n+1 of trigonometric polynomials of degree n. Notice also that a flex of order one, i.e. the case n = 0, is nothing but a critical point. The existence of 2n + 2 flexes of order 2n + 1 for any C 2n+1 -function u on S 1 is an easy consequence of the well-known fact that a function has at least 2n + 2 zeros if its Fourier coefficients a i and b i vanish for i ≤ n;
Abstract. For a real valued periodic smooth function u on R, n ≥ 0, one defines the osculating polynomial ϕs (of order 2n + 1) at a point s ∈ R to be the unique trigonometric polynomial of degree n, whose value and first 2n derivatives at s coincide with those of u at s. We will say that a point s is a clean maximal flex (resp. clean minimal flex ) of the function u on S 1 if and only if ϕs ≥ u (resp. ϕs ≤ u) and the preimage (ϕ − u) −1 (0) is connected. We prove that any smooth periodic function u has at least n + 1 clean maximal flexes of order 2n + 1 and at least n + 1 clean minimal flexes of order 2n + 1. The assertion is clearly reminiscent of Morse theory and generalizes the classical four vertex theorem for convex plane curves. §1. Introduction For a real valued C 2n -function u on S 1 = R/2πZ, n ≥ 0, one defines the osculating polynomial ϕ s (of order 2n + 1) at a point s ∈ S 1 to be the unique trigonometric polynomial of degree n, ϕ s (t) = a 0 + a 1 cos t + b 1 sin t + · · · + a n cos nt + b n sin nt, whose value and first 2n derivatives at s coincide with those of u at s. If u is C 2n+1 and the value and the first 2n + 1 derivatives of u and ϕ s coincide in s, i.e., if ϕ s hyperosculates u in s, then we call s a flex of u (of order 2n+1). Notice that the order 2n + 1 of the osculating polynomials and flexes in the definition above has been chosen such that it coincides with the dimension of the space A 2n+1 of trigonometric polynomials of degree n. Notice also that a flex of order one, i.e. the case n = 0, is nothing but a critical point. The existence of 2n + 2 flexes of order 2n + 1 for any C 2n+1 -function u on S 1 is an easy consequence of the well-known fact that a function has at least 2n + 2 zeros if its Fourier coefficients a i and b i vanish for i ≤ n;
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