A set of orthogonal polynomials induced by a given orthogonal polynomial

Abstract

Dedicated to the memory of Alexander M. Ostrowski on the occasion of the lOOth anniversary of his birth.Summary. Given an integer n/> 1, and the orthogonal polynomials n,( -; dcr) of degree n relative to some positive measure da, the polynomial system "induced" by n, is the system of orthogonal polynomials {r~k. ~ } corresponding to the modified measure d~, = n~ d~. Our interest here is in the problem of determining the coefficients in the three-term recurrence relation for the polynomials ~k,, from the recursion coefficients of the orthogonal polynomials belonging to the measure do. A stable computational algorithm is proposed, which uses a sequence of QR steps with shifts. For all four Chebyshev measures da, the desired coefficients can be obtained analytically in closed form. For Chebyshev measures of the first two kinds this was shown by AI-Salam, Allaway and Askey, who used sieved orthogonal polynomials, and by Van Assche and Magnus via polynomial transformations. Here, analogous results are obtained by elementary methods for Chebyshev measures of the third and fourth kinds. (The same methods are also applicable to the other two Chebyshev measures.) Interlacing properties involving the zeros of n, and those of ~, + ~,, are studied for Gegenbauer measures, as well as the orthogonality--or lack thereof--of the polynomial sequence {~ .... i }.

On rational transformations of linear functionals: direct problem

“…(3.6) Notice that 1 2 R n (x) are the monic Chebyshev polynomials of fourth kind, that is the monic Jacobi polynomials with parameters α = 1/2 and β = −1/2, see [5].…”

The error bounds of Gauss quadrature formulae for the modified weight functions of Chebyshev type

“…and using again (2.7), the proof of (2.3) is fulfilled. [4,Theorem 3.4]), whereŮ n denotes the monic Chebyshev polynomial of the second kind, and the kernel is given by K [2] n (z) = ̺ [2] n (z) π [2] n (z)…”

“…It is well known that the zeros and nodes of the Gauss rule can be efficiently computed by means of the eigenvalues and eigenvectors of the related tridiagonal Jacobi matrix. In general, it is not feasible to get closed analytic expressions of the entries of the Jacobi matrix for the induced measure d σ n in terms of the corresponding for dσ; in this sense, in [4] a stable numerical algorithm is given. But in the particular case of the well-known four Chebyshev weights dσ [i] , i = 1, 2, 3, 4 , the related induced orthogonal polynomials { π [i] m,n } are easily expressible as combinations of Chebyshev polynomials of the first kind T k (i.e., orthogonal polynomials with respect to the Chebyshev weight dσ [1] (see [4, §3]).…”

Orthogonal Polynomials for Modified Chebyshev Measure of the First Kind

“…In [3], Gautschi and Li considered the special case s = 1 and proved the following result, which will be used in the proof of our main result as a base of induction.…”