Centrally symmetric orthogonal polynomials and second order partial differential equations
نویسندگان
چکیده
منابع مشابه
Sobolev Orthogonal Polynomials in Two Variables and Second Order Partial Differential Equations
We consider polynomials in two variables which satisfy an admissible second order partial differential equation of the form (*) Auxx + 2Buxy + Cuyy +Dux + Euy = u; and are orthogonal relative to a symmetric bilinear form de ned by '(p; q) = h ; pqi+ h ; pxqxi ; where A; ; E are polynomials in x and y; is an eigenvalue parameter, and are linear functionals on polynomials. We nd a condition for ...
متن کاملOrthogonal Polynomials and Partial Differential Equations on the Unit Ball
Orthogonal polynomials of degree n with respect to the weight function Wμ(x) = (1 − ‖x‖2)μ on the unit ball in R are known to satisfy the partial differential equation [ ∆− 〈x,∇〉 − (2μ+ d)〈x,∇〉 ] P = −n(n+ 2μ+ d)P for μ > −1. The singular case of μ = −1,−2, . . . is studied in this paper. Explicit polynomial solutions are constructed and the equation for ν = −2,−3, . . . is shown to have comple...
متن کاملOrthogonal matrix polynomials satisfying second order difference equations
We develop a method that allows us to construct families of orthogonal matrix polynomials of size N ×N satisfying second order difference equations with polynomial coefficients. The existence (and properties) of these orthogonal families strongly depends on the non commutativity of the matrix product, the existence of singular matrices and the matrix size N .
متن کاملOrthogonal Polynomials on the Unit Ball and Fourth-Order Partial Differential Equations
The purpose of this work is to analyse a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which includes an additional term on the sphere. First, we will get connection formulas relating classical multivariate orthogonal polynomials on the ball with our family of orthogonal polynomials. Then, using the representation of these polynomials in terms of sp...
متن کاملOrthogonal Polynomial Eigenfunctions of Second-order Partial Differerential Equations
In this paper, we show that for several second-order partial differential equations L[u] = A(x, y)uxx + 2B(x, y)uxy + C(x, y)uyy +D(x, y)ux + E(x, y)uy = λnu which have orthogonal polynomial eigenfunctions, these polynomials can be expressed as a product of two classical orthogonal polynomials in one variable. This is important since, otherwise, it is very difficult to explicitly find formulas ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Methods and Applications of Analysis
سال: 2000
ISSN: 1073-2772,1945-0001
DOI: 10.4310/maa.2000.v7.n1.a3