On Recursively Defined Orthogonal Polynomials 1
نویسنده
چکیده
ra + n)» T(0x r(£ + v)x F« + n)xk + r(f + t,)x* T(£)xk + T(Qxk T(Qx = T(t + r, h)[T{to)x F(£o)xt] + [r(f + n)** r(ö*J + T({ to)[TQ;0)xt rft,)*] G r({ + t, £0)F2 + F2 + F(£ ¿0)7, C Fi + Vi + Vi C V; i.e., [m+7])-T(0]BCVlor |t?| <Ô, ag£+77go, ÇaF(£). That F(£) is compact for £>£o follows from the fact F(£0) is a compact operator and F(£) = F(^ — ̂ o)F(^0) where F(£—f0) is a continuous linear map of £ into £.
منابع مشابه
Recurrence Formulas for Multivariate Orthogonal Polynomials
In this paper, necessary and sufficient conditions are given so that multivariate orthogonal polynomials can be generated by a recurrence formula. As a consequence, orthogonal polynomials of total degree n in d variables that have dim n¡( common zeros can now be constructed recursively. The result is important to the construction of Gaussian cubature formulas.
متن کاملRakhmanov'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...
متن کاملSolving singular integral equations by using orthogonal polynomials
In this paper, a special technique is studied by using the orthogonal Chebyshev polynomials to get approximate solutions for singular and hyper-singular integral equations of the first kind. A singular integral equation is converted to a system of algebraic equations based on using special properties of Chebyshev series. The error bounds are also stated for the regular part of approximate solut...
متن کاملA New Class of Orthogonal Polynomials1
A new class of orthogonal polynomials is introduced which generalizes the Bernstein-Szegö polynomials and includes the associated polynomials as well. The purpose of this paper is to give a natural extension of the Bernstein-Szegö orthogonal polynomials for a general class of weight functions. A nonnegative function w defined on the real line is called a weight function if w > 0, fRw > 0 and al...
متن کاملLogarithmic behavior of some combinatorial sequences
Two general methods for establishing the logarithmic behavior of recursively defined sequences of real numbers are presented. One is the interlacing method, and the other one is based on calculus. Both methods are used to prove logarithmic behavior of some combinatorially relevant sequences, such as Motzkin and Schröder numbers, sequences of values of some classic orthogonal polynomials, and ma...
متن کاملMultiple Orthogonal Polynomials on the Semicircle and Corresponding Quadratures of Gaussian Type1
In this paper multiple orthogonal polynomials defined using orthogonality conditions spread out over r different measures are considered. We study multiple orthogonal polynomials on the real line, as well as on the semicircle (complex polynomials orthogonal with respect to the complex-valued inner products (f, g)k = ∫ π 0 f(e)g(e)wk(e ) dθ, for k = 1, 2, . . . , r). For r = 1, in the real case ...
متن کامل