The Mean Convergence of Orthogonal Series of Polynomials

The Mean Convergence of Orthogonal Series of Polynomials.

2 9 Ja n 20 04 MEAN CONVERGENCE OF ORTHOGONAL FOURIER SERIES AND INTERPOLATING POLYNOMIALS

For a family of weight functions that include the general Jacobi weight functions as special cases, exact condition for the convergence of the Fourier orthogonal series in the weighted L space is given. The result is then used to establish a Marcinkiewicz-Zygmund type inequality and to study weighted mean convergence of various interpolating polynomials based on the zeros of the corresponding o...

Weak Convergence of Orthogonal Polynomials

The weak convergence of orthogonal polynomials is given under conditions on the asymptotic behaviour of the coefficients in the three-term recurrence relation. The results generalize known results and are applied to several systems of orthogonal polynomials, including orthogonal polynomials on a finite set of points.

Fourier Series of Orthogonal Polynomials

It follows from Bateman [4] page 213 after setting = 1 2 . It can also be found with slight modi cation in Bateman [5] page122. However we are not aware of any reference where explicit formulas for the Fourier coef cients for Gegenbauer, Jacobi, Laguerre and Hermite polynomials can be found. In this article we use known formulas for the connection coef cients relating an arbitrary orthogonal po...

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