Orthonormal Polynomials under Different Inner Product Measures

This Demonstration shows how the orthonormal polynomials of a space where the inner product is a weighted integral vary with the weight function , . Here, the monomial basis is orthonormalized by the Gram–Schmidt process with respect to the inner product and the weight functions are probability measures (so that ) corresponding to the Legendre, Hermite, Laguerre and generalized Laguerre polynomials.


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The variable is often time [1].
[1] A. Gu, T. Dao, S. Ermon, A. Rudra and C. Re, "Hippo: Recurrent Memory with Optimal Polynomial Projections," Advances in Neural Information Processing Systems, 33, 2020 pp. 1474–1487. par.nsf.gov/biblio/10214620.
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