Fourier summation with kernels defined by Jacobi polynomials
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Trigonometric polynomial kernels defined by Jacobi polynomials are investigated. They generalize the classical Dirichlet kernel and the Fejer kernel. The asymptotic behavior of the corresponding Fourier summation obtained is leading to optimal kernels. The Dirichlet kernel and the Fejer kernel are basic tools in the theory of classical Fourier expansions. We introduce trigonometric polynomial kernels defined by Jacobi polynomials. One class consists of kernels with alternating signs containing the Dirichlet kernel as a special case; the other class consists of nonnegative kernels containing the Fejfr kernel as a special case. Therefore, these kernels are certain parameterized generalizations of the Dirichlet kernel resp. Fejer kernel. We investigate their asymptotic behavior, finding in this way optimal kernels, where optimality is defined with respect to the generalized Lipschitz condition. Throughout this work our main reference for notation and results concerning Fourier series is [4]. For other uses of Jacobi polynomials as summability kernels we refer to [2, 6]. 1. GENERALIZATIONS OF THE DIRICHLET KERNEL Fix A > 0. We study the following row-finite 0 factors, which are strongly related to certain Jacobi polynomials: For n E N0, k = 0, ... , n, set (1 ) A _ (k)n+k(A)n-k(n!)2 and aA _A an,k -(n + k)! (n k)! ((A~)2 an -k n,k and for t e] a, r], denote n A. ikt (2) DA(eit) = aflke k=-n For A = 1 we have the classical Dirichlet kernel. Proposition 1. We have (3) DA(eit) = bARnA1 112)(cos t), Received by the editors May 30, 1990 and, in revised form, October 1, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 42A10.
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