Orthogonal polynomials by Gabor Szego PDF

Orthogonal polynomials by Gabor Szego PDF

By Gabor Szego

Fresh years have visible loads of development within the box of orthogonal polynomials, a subject matter heavily with regards to many vital branches of study. Orthogonal polynomials are hooked up with trigonometric, hypergeometric, Bessel, and elliptic services, are regarding the idea of persevered fractions and to special difficulties of interpolation and mechanical quadrature, and are of occasional incidence within the theories of differential and indispensable equations. moreover, they provide relatively basic and instructive illustrations of yes events within the conception of orthogonal platforms. lately, a few of these polynomials were proven to be of value in quantum mechanics and in mathematical statistics.

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11) is due to Pinkus [45]. In view of the difficulty of identifying optimal sample points it is comforting to know that much less effort is required to show that equally spaced points are asymptotically optimal for any r > 1, p > q. We now present the p r o o f of this result. 15) inf a sup Ilf - a ( f ( 1 ..... f ( ~ l n+ 1 )) Ilq -< d . 15), is easily proved by using a local polynomial interpolation scheme. Thus we divide [0,1 ] into m equally spaced intervals. On each of these intervals we interpolate f at r equally spaced points by a polynomial of degree at most r - l .

Since our algorithm clearly produces a function in Lq[0,1] we do not pursue these alternatives further here. F o r the lower b o u n d we follow the m e t h o d used in M a k o z o v [25]. Let ~ be any C ~ function with support on (0,1) n o r m a l i z e d so that [t q~(r) [I p = 1 . C h o o s e any 58 t = ( t l , . . 16) E Itil = 1. ,n + 1 ; x 0 =0. Let I be any continuous map from Wp[O,1] into IR n . Then t -,- I ( f ( o , t ) ) is an odd continuous map and so by the Borsuk antipodality t h e o r e m (cf.

N 40 and N 2 aiq(xi) = 0 , i=l q E ~ m _ l ( I R n) . Details in D u c h o n [10] and Meinguet [26]. 1. 4. A ( M e l k m a n [27]) X = {f e L2(IR) : f(~o) = 0, I¢o I > o} where f denotes the Fourier transform of f. Thus X is the space of b a n d - l i m i t e d functions. Let Y = IRn and If = (f(t I) ..... f(tn)), then n I * ( a l ..... ,n, ~r(t - ti) a w e l l - k n o w n procedure. Next we turn to a result on the existence of linear optimal algorithms in the presence of inaccurate i n f o r m a t i o n (e > 0).

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