Orthogonal matrix polynomials and quadrature formulas
نویسندگان
چکیده
منابع مشابه
Construction of σ-orthogonal Polynomials and Gaussian Quadrature Formulas
Let dα be a measure on R and let σ = (m1,m2, ..., mn), where mk ≥ 1, k = 1, 2, ..., n, are arbitrary real numbers. A polynomial ωn(x) := (x − x1)(x − x2)...(x − xn) with x1 ≤ x2 ≤ ... ≤ xn is said to be the n-th σ-orthogonal polynomial with respect to dα if the vector of zeros (x1, x2, ..., xn) is a solution of the extremal problem ∫
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Let pk(x) = x +■■■ , k e N0 , be the polynomials orthogonal on [-1, +1] with respect to the positive measure da . We give sufficient conditions on the real numbers p , j = 0, ... , m , such that the linear combination of orthogonal polynomials YfLo^jPn-j has n simple zeros in (—1,-1-1) and that the interpolatory quadrature formula whose nodes are the zeros of Yfj=oßjPn-j has positive weights.
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By using the three-term recurrence equation satisfied by a family of orthogonal polynomials, the Christoffel-Darboux-type bilinear generating function and their asymptotic expressions, we obtain quadrature formulas for integral transforms generated by the classical orthogonal polynomials. These integral transforms, related to the so-called Poisson integrals, correspond to a modified Fourier Tra...
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Various concepts of orthogonality on the real line are reviewed that arise in connection with quadrature rules. Orthogonality relative to a positive measure and Gauss-type quadrature rules are classical. More recent types of orthogonality include orthogonality relative to a sign-variable measure, which arises in connection with Gauss-Kronrod quadrature, and power (or implicit) orthogonality enc...
متن کاملOrthogonal matrix polynomials, scalar-type Rodrigues' formulas and Pearson equations
Some families of orthogonal matrix polynomials satisfying second order differential equations with coefficients independent of n have recently been introduced (see [DG1]). An important difference with the scalar classical families of Jacobi, Laguerre and Hermite, is that these matrix families do not satisfy scalar type Rodrigues’ formulas of the type (ΦnW )W, where Φ is a matrix polynomial of d...
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ژورنال
عنوان ژورنال: Linear Algebra and its Applications
سال: 2002
ISSN: 0024-3795
DOI: 10.1016/s0024-3795(01)00474-8