On a filter for exponentially localized kernels based on Jacobi polynomials

Let @a,@b>=-12, and for k=0,1,..., p"k^(^@a^,^@b^) denote the orthonormalized Jacobi polynomial of degree k. We discuss the construction of a matrix H so that there exist positive constants c, c"1, depending only on H, @a, and @b such that @?k=0~H"k","np"k^(^@a^,^@b^)(cos@q)p"k^(^@a^,^@b^)(cos@f)@?c"1n^2^m^a^x^(^@a^,^@b^)^+^2exp(-cn(@q-@f)^2),@q,@f@?[0,@p],n=1,2,....Specializing to the case of Chebyshev polynomials, @a=@b=-12, we apply this theory to obtain a construction of an exponentially localized polynomial basis for the corresponding L^2 space.

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