Abstract
The connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many applications. While the connection problem may be solved by directly computing the change–of–basis matrix, this approach is computationally expensive. A recent approach to solving the connection problem involves the use of the spectral connection matrix, which is a matrix whose eigenvector matrix is the desired change–of–basis matrix. In Bella and Reis (2014), it is shown that for the connection problem between any two different classical real orthogonal polynomials of the Hermite, Laguerre, and Gegenbauer families, the related spectral connection matrix has quasiseparable structure. This result is limited to the case where both the source and target families are one of the Hermite, Laguerre, or Gegenbauer families, which are each defined by at most a single parameter. In particular, this excludes the large and common class of Jacobi polynomials, defined by two parameters, both as a source and as a target family. In this paper, we continue the study of the spectral connection matrix for connections between real orthogonal polynomial families. In particular, for the connection problem between any two families of the Hermite, Laguerre, or Jacobi type (including Chebyshev, Legendre, and Gegenbauer), we prove that the spectral connection matrix has quasiseparable structure. In addition, our results also show the quasiseparable structure of the spectral connection matrix from the Bessel polynomials, which are orthogonal on the unit circle, to any of the Hermite, Laguerre, and Jacobi types. Additionally, the generators of the spectral connection matrix are provided explicitly for each of these cases, allowing a fast algorithm to be implemented following that in Bella and Reis (2014).
Highlights
Let {Pk (x)}∞k=0 be a sequence of real–valued polynomials, with deg(Pk (x)) = k, and let w(x) be a non–negative real–valued weight function on some interval [a, b]
Our results show the quasiseparable structure of the spectral connection matrix from the Bessel polynomials, which are orthogonal on the unit circle, to any of the Hermite, Laguerre, and Jacobi types
At the conclusion will we discuss a connection to the Bessel polynomials, which are often considered classical they are orthogonal on the unit circle
Summary
The authors propose an algorithm for producing recurrence relations for desired connection coefficients and provide some examples While their method uses only the recurrence relations for the orthogonal families and covers a very wide range of cases, the authors readily acknowledge the shortcomings of their approach by stating that they “do not succeed in resolving it through a compact form." They continue, explaining that “it is required to guess a closed form for the solution of the recurrence from enough data produced by a symbolic programming language [16].”. We continue the work in [24] by proving that the spectral connection matrix has quasiseparable rank structure when the target family is the large and very useful family of Jacobi polynomials, including Chebychev polynomials as special cases. A connection to the Bessel polynomials is made in Section 6, and some conclusions are offered in the final section
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