Orthogonal Polynomials TCU Seminar Lecture Notes
نویسنده
چکیده
And a disclaimer: I've made lots of changes of variables throughout. Expect some mistakes. I would appreciate hearing about any you find. I want to look at two different topics that have to do with orthogonal polynomials. The first has to do with approximation; the second, with Mellin transforms and zeta functions. If we have an inner product on R[x], we can use Gram-Schmidt to convert {1, x, x 2 ,. . .} into an orthogonal basis of monic polynomials {p n (x)}. Our inner products will have the form p, q = b a p(x) q(x) w(x) dx for some weight function w. A family of orthogonal polynomials will have p n of degree n, but not necessarily monic. For a given weight function, we may always multiply each polynomial by an arbitrary constant to get another family. Standard choices are monic, normalized to p n , p n = 1, or to coincide with nice generating functions.
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