Goh, Say Song; Goodman, Tim N. T.; Lee, S. L. Orthogonal polynomials, biorthogonal polynomials and spline functions. (English) Zbl 1460.41004 Appl. Comput. Harmon. Anal. 52, 141-164 (2021). For orthogonal and bi-orthogonal polynomials, there are several important concepts that the authors of this article succeed to generalise. Among them are so-called generating functions and the Rodrigues’ formula. Concretely, the authors wish to replace the expressions \(f_m\) in the Rodrigues’ formula which amount to weight-functions \(\omega\) times some simple powers (often, monomials or even constants, or at most quadratics anyway, that are then taken to \(m\)-th powers) in the classical cases, by B-splines. The \(f_m\) are assumed to be decaying exponentially, as \(O(\exp(-Bx-\epsilon x))\). As a result, their Fourier transforms become analytic in an open strip of size \(B\).For comparison, for Laguerre polynomials for example, the \(f_m\) would be multiples of an exponential weight function times \(x^m\), for Hermite weight function \(\exp(-x^2)\) (times \(1^m\)), and these are now replaced by B-splines.Both uniform B-splines (pen-ultimate section) and non-uniform knots (last section of the article) are treated, and there is even a further generalisation where the B-splines are defined by repeated convolution not starting from a characteristic function of an interval (the classical case), but a less restricted function as an initial value. The Fourier transforms of those “generalised B-Splines” alter in that the usual powers of sinc-functions get another factor of the Fourier transform of the starting (initial) function.The result is called a generalised Rodrigues’ formula and the needed generating function is called a generalised generating function. In order to form the generalised generating function, Fourier transforms for the \(f_m\) are needed which would be simply convolutions of weight functions with some derivatives of \(\delta\)-functions in the classical case (i.e., derivatives of the weight function). This reformulated Rodrigues’ formula \(\mu_m=(-1)^m f_m^{(m)}\) is equivalent to considering the \(\mu_m\)s as bi-orthogonal to a sequence of orthogonal polynomials \(Q_m\) that are one factor of the expansion coefficients of the generalised generating function; the other factor being the Fourier transforms of the \(f_m\): \[\exp(xz)=\sum\nolimits_0^\infty Q_m(x) z^m \hat f_m(\mathrm{i}z).\] The main results guarantee the existence of the \(Q_m\) polynomials and the well-definedness of the generalised generating function (these results are, in particular, Theorem 2.3 and Theorem 3.1 and – for the most general case – Theorem 4.3). Reviewer: Martin D. Buhmann (Gießen) Cited in 2 Documents MSC: 41A15 Spline approximation 41A30 Approximation by other special function classes 41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 65D07 Numerical computation using splines Keywords:orthogonal polynomials; biorthogonal polynomials and spline functions; generalized generating functions; generalized Rodrigues formula; uniform \(B\)-splines; refinable functions; nonuniform \(B\)-splines Software:DLMF PDFBibTeX XMLCite \textit{S. S. Goh} et al., Appl. Comput. Harmon. Anal. 52, 141--164 (2021; Zbl 1460.41004) Full Text: DOI References: [1] de Boor, C., A Practical Guide to Splines (1978), Springer-Verlag: Springer-Verlag New York · Zbl 0406.41003 [2] Chui, C. 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