An Equivalence Theorem for Series of Orthogonal Polynomials

An Equivalence Theorem for Series of Orthogonal Polynomials.

xj(u, v) = yj(u, v), j = 1, 2, 3, so that again (8) holds. Hence, under our hypotheses, D is mapped isothermically on a spherical surface of finite radius, and circles are not mapped on circles. III. Characterization of Those Isothermic Spherical Maps Which Map Circles on Circles and of Isothermic Maps on Minimal Surfaces.-THEOREM 3. If thefunctions (6) have continuous partial derivatives of th...

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...

Blumenthal's Theorem for Laurent Orthogonal Polynomials

We investigate polynomials satisfying a three-term recurrence relation of the form Bn(x) = (x − βn)Bn−1(x) − αnxBn−2(x), with positive recurrence coefficients αn+1, βn (n = 1, 2, . . .). We show that the zeros are eigenvalues of a structured Hessenberg matrix and give the left and right eigenvectors of this matrix, from which we deduce Laurent orthogonality and the Gaussian quadrature formula. ...

On the continuous analog of Rakhmanov's theorem for orthogonal polynomials

Rakhmanov's theorem for orthogonal matrix polynomials on the unit circle

Rakhmanov’s theorem for orthogonal polynomials on the unit circle gives a sufficient condition on the orthogonality measure for orthogonal polynomials on the unit circle, in order that the reflection coefficients (the recurrence coefficients in the Szegő recurrence relation) converge to zero. In this paper we give the analog for orthogonal matrix polynomials on the unit circle. 1. Rakhmanov’s t...