Divergent Legendre-sobolev Polynomial Series
نویسنده
چکیده
Let be introduced the Sobolev-type inner product (f, g) = 1 2 Z 1 −1 f(x)g(x)dx + M [f ′(1)g′(1) + f ′(−1)g′(−1)], where M ≥ 0. In this paper we will prove that for 1 ≤ p ≤ 4 3 there are functions f ∈ L([−1, 1]) whose Fourier expansion in terms of the orthonormal polynomials with respect to the above Sobolev inner product are divergent almost everywhere on [−1, 1]. We also show that, for some values of δ, there are functions whose Legendre-Sobolev expansions have almost everywhere divergent Cesàro means of order δ. AMS Mathematics Subject Classification (2000): 42C05, 42C10
منابع مشابه
Cubature over the sphere S in Sobolev spaces of arbitrary order
This paper studies numerical integration (or cubature) over the unit sphere S2 ⊂ R3 for functions in arbitrary Sobolev spaces Hs(S2), s > 1. We discuss sequences (Qm(n))n∈N of cubature rules, where (i) the rule Qm(n) uses m(n) points and is assumed to integrate exactly all (spherical) polynomials of degree ≤ n, and (ii) the sequence (Qm(n)) satisfies a certain local regularity property. This lo...
متن کاملOn Polar Legendre Polynomials
We introduce a new class of polynomials {Pn}, that we call polar Legendre polynomials, they appear as solutions of an inverse Gauss problem of equilibrium position of a field of forces with n + 1 unit masses. We study algebraic, differential and asymptotic properties of this class of polynomials, that are simultaneously orthogonal with respect to a differential operator and a discrete-continuou...
متن کاملCubature over the sphere S2 in Sobolev spaces of arbitrary order
This paper studies numerical integration (or cubature) over the unit sphere S2 ⊂ R3 for functions in arbitrary Sobolev spaces Hs(S2), s > 1. We discuss sequences (Qm(n))n∈N of cubature rules, where (i) the rule Qm(n) uses m(n) points and is assumed to integrate exactly all (spherical) polynomials of degree n and (ii) the sequence (Qm(n)) satisfies a certain local regularity property. This local...
متن کاملA polynomial approximation for arbitrary functions
Abstract We describe an expansion of Legendre polynomials, analogous to the Taylor expansion, to approximate arbitrary functions. We show that the polynomial coefficients in Legendre expansion, thus, the whole series, converge to zero much more rapidly compared to the Taylor expansion of the same order. Furthermore, using numerical analysis with sixth-order polynomial expansion, we demonstrate ...
متن کاملDivergent Cesàro Means of Jacobi-Sobolev Expansions
Let μ be the Jacobi measure supported on the interval [−1, 1]. Let introduce the Sobolev-type inner product 〈f, g〉 = ∫ 1 −1 f(x)g(x) dμ(x) +Mf(1)g(1) +Nf ′(1)g′(1), where M,N ≥ 0. In this paper we prove that, for certain indices δ, there are functions whose Cesàro means of order δ in the Fourier expansion in terms of the orthonormal polynomials associated with the above Sobolev inner product ar...
متن کامل