An extremal property of Hermite polynomials |
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Authors: | Geno Nikolov |
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Institution: | Department of Mathematics, University of Sofia, 5 James Bourchier Boulevard, 1164, Sofia, Bulgaria |
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Abstract: | Let Hn be the nth Hermite polynomial, i.e., the nth orthogonal on
polynomial with respect to the weight w(x)=exp(−x2). We prove the following: If f is an arbitrary polynomial of degree at most n, such that |f||Hn| at the zeros of Hn+1, then for k=1,…,n we have f(k)Hn(k), where · is the
norm. This result can be viewed as an inequality of the Duffin and Schaeffer type. As corollaries, we obtain a Markov-type inequality in the
norm, and estimates for the expansion coefficients in the basis of Hermite polynomials. |
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Keywords: | Gauss-type quadrature formulae Hermite polynomials Laguerre polynomials Duffin- and Schaeffer-type inequality |
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