Matrix Calculus-Based Approach to Orthogonal Polynomial Sequences
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Matrix Calculus-Based Approach to Orthogonal Polynomial Sequences F. A. Costabile, M. I. Gualtieri and A. Napoli Abstract. In this paper, an approach to orthogonal polynomials based on matrix calculus is proposed. Known and new basic results are given, such as recurrence relations and determinant forms. New algorithms, similar, but not identical, to the Chebyshev one, for practical calculation of the polynomials are presented. The cases of monic and symmetric orthogonal polynomial sequences and the case of orthonormal polynomial sequences have been considered. Some classical and non-classical examples are given. The work is framed in a broader perspective, already started by the authors. It provides for the determination of properties of a general sequence of polynomials and, therefore, their applicability to special classes of the most important polynomials. Mathematics Subject Classification. 42C05, 11B83. Keywords. Orthogonal polynomials, Polynomial sequences, Determinant forms.
1. Introduction Orthogonal polynomial systems play an important role in approximation theory, in mathematical and numerical analysis and applications (for example, Gaussian quadrature, least square approximation of functions, and differential and difference equations). There is a wide literature on this subject, starting from the first work of Adrien–Marie Legendre [34] on the gravitational potential in spherical coordinates, up to the present day (see, for example, [1–3,7,15,28,31–33, 35–37,39,42–46] and references therein). Particularly, Gautschi [17–23,25– 27] developed the so-called constructive theory of orthogonal polynomials on IR, including effective algorithms for numerically generating orthogonal polynomials. Let P be the space of polynomials in one variable with real coefficients. For every n ∈ IN, let Pn ⊂ P be the space of polynomials of degree ≤ n. Let 0123456789().: V,-vol
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dμ(t) be a positive measure on IR with bounded or unbounded support for which all the moments: μk = tk dμ(t), μ0 > 0, (1.1) IR
exist and are finite. Then, the inner product: p(t)q(t)dμ(t) (p, q) =
(1.2)
IR
is well defined∀p,q ∈ P. There exists a unique sequence of monic orthogonal polynomials, qn n∈IN , being qn = qn (t) a polynomial of degree n: qn (t) = qn (t, dμ(t)) = tn + terms of lower degree and (qk , qn ) = ||qn ||2 δk,n , where
||qn ||2 = IR
qn2 (t)dμ(t).
The Gautschi’s constructive theory is essentially based on the threeterm recurrence relation: (1.3) qn+1 (t) = (t − αn )qn (t) − βn qn−1 (t), n = 0, 1, . . . with q−1 (t) = 0, q0 (t) = 1, and αn n∈IN , βn n∈IN numerical sequences depending on the measure dμ(t). Theparameter β0 may be arbitrary, but it is conveniently defined by β0 = μ0 = IR dμ(t), and βn > 0, ∀n ≥ 1. The coefficients αn , βn in (1.3) are fundamental quantities in the constructive theory of orthogonal polynomials [18]. In fact, they provide a compact way of representing and easily calculating orthogonal polynomials, their derivatives, their zeros, and Gaussian coeffi
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