The oscillation of differential transforms - entire Jacobi expansions |
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Authors: | C.L. Prather |
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Affiliation: | Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA |
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Abstract: | Let L=(1−x2)D2−((β−α)−(α+β+2)x)D with , and . Let f∈C∞[−1,1], , with normalized Jacobi polynomials and the Cn decrease sufficiently fast. Set Lk=L(Lk−1), k?2. Let ρ>1. If the number of sign changes of (Lkf)(x) in (−1,1) is O(k1/(ρ+1)), then f extends to be an entire function of logarithmic order . For Legendre expansions, the result holds with replaced with . |
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Keywords: | Sign changes Iterated operator Entire function |
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