By L. Reichel, W. Gautschi, F. Marcellan
Orthogonal polynomials play a favourite function in natural, utilized, and computational arithmetic, in addition to within the technologies. it's the target of the current quantity within the sequence "Numerical research within the twentieth Century" to study, and occasionally expand, a number of the many identified effects and homes of orthogonal polynomials and similar quadrature ideas. additionally, this quantity discusses ideas to be had for the research of orthogonal polynomials and linked quadrature principles. certainly, the layout and computation of numerical integration equipment is a crucial zone in numerical research, and orthogonal polynomials play a primary position within the research of many integration methods.
The twentieth century has witnessed a fast improvement of orthogonal polynomials and comparable quadrature principles, and we for that reason can't even try to assessment all major advancements inside this quantity. We basically have sought to stress effects and methods which have been of value in computational or utilized arithmetic, or which we think could lead on to major growth in those components within the close to destiny. regrettably, we can't declare completeness even inside this constrained scope. however, we are hoping that the readers of the amount will locate the papers of curiosity and lots of references to comparable paintings of help.
We define the contributions within the current quantity. homes of orthogonal polynomials are the focal point of the papers by means of Marcellán and Álvarez-Nodarse and through Freund. the previous contribution discusses "Favard's theorem", i.e., the query lower than which stipulations the recurrence coefficients of a kin of polynomials confirm a degree with admire to which the polynomials during this kinfolk are orthogonal. Polynomials that fulfill a three-term recurrence relation in addition to Szegõ polynomials are thought of. The degree is permitted to be signed, i.e., the instant matrix is authorized to be indefinite. Freund discusses matrix-valued polynomials which are orthogonal with recognize to a degree that defines a bilinear shape. This contribution makes a speciality of breakdowns of the recurrence kin and discusses suggestions for overcoming this trouble. Matrix-valued orthogonal polynomials shape the root for algorithms for reduced-order modeling. Freund's contribution to this quantity presents references to such algorithms and their program to circuit simulation.
The contribution by way of Peherstorfer and Steinbauer analyzes inverse pictures of polynomial mappings within the complicated aircraft and their relevance to extremal homes of polynomials orthogonal with appreciate to measures supported on quite a few units, reminiscent of a number of durations, lemniscates, or equipotential strains. functions comprise fractal thought and Julia etc.
Orthogonality with admire to Sobolev internal items has attracted the curiosity of many researchers over the last decade. The paper by means of Martinez discusses many of the fresh advancements during this zone. The contribution by way of López Lagomasino, Pijeira, and Perez Izquierdo offers with orthogonal polynomials linked to measures supported on compact subsets of the advanced airplane. the site and asymptotic distribution of the zeros of the orthogonal polynomials, in addition to the nth-root asymptotic habit of those polynomials is analyzed, utilizing tools of capability theory.
Investigations in line with spectral concept for symmetric operators delivers perception into the analytic homes of either orthogonal polynomials and the linked Padé approximants. The contribution by way of Beckermann surveys those results.
Van Assche and Coussement examine a number of orthogonal polynomials. those polynomials come up in simultaneous rational approximation; particularly, they shape the basis for simultaneous Hermite-Padé approximation of a process of numerous services. The paper compares a number of orthogonal polynomials with the classical households of orthogonal polynomials, resembling Hermite, Laguerre, Jacobi, and Bessel polynomials, utilizing characterization theorems.
Bultheel, González-Vera, Hendriksen, and Njåstad think of orthogonal rational features with prescribed poles, and talk about quadrature ideas for his or her distinct integration. those quadrature ideas could be seen as extensions of quadrature ideas for Szegõ polynomials. The latter ideas are unique for rational features with poles on the beginning and at infinity.
Many of the papers of this quantity are taken with quadrature or cubature principles concerning orthogonal polynomials. The research of multi variable orthogonal polynomials types the basis of many cubature formulation. The contribution of Cools, Mysovskikh, and Schmid discusses the relationship among cubature formulation and orthogonal polynomials. The paper stories the improvement initiated by way of Radon's seminal contribution from 1948 and discusses open questions. The paintings through Xu offers with multivariate orthogonal polynomials and cubature formulation for a number of areas in Rd. Xu exhibits that orthogonal constructions and cubature formulation for those areas are heavily related.
The paper by way of Milovanovic bargains with the homes of quadrature ideas with a number of nodes. those principles generalize the Gauss-Turán principles. Moment-preserving approximation through faulty splines is taken into account as an application.
Computational concerns with regards to Gauss quadrature principles are the subject of the contributions by way of Ehrich and Laurie. The latter paper discusses numerical equipment for the computation of the nodes and weights of Gauss-type quadrature ideas, while moments, changed moments, or the recursion coefficients of the orthogonal polynomials linked to a nonnegative degree are identified. Ehrich is worried with easy methods to estimate the mistake of quadrature ideas of Gauss kind. this query is necessary, e.g., for the layout of adaptive quadrature exercises in response to ideas of Gauss type.
The contribution via Mori and Sugihara experiences the double exponential transformation in numerical integration and in a number of Sinc tools. this variation allows effective evaluate of the integrals of analytic services with endpoint singularities.
Many algorithms for the answer of large-scale difficulties in technological know-how and engineering are in response to orthogonal polynomials and Gauss-type quadrature principles. Calvetti, Morigi, Reichel, and Sgallari describe an software of Gauss quadrature to the computation of bounds or estimates of the Euclidean norm of the mistake in iterates (approximate suggestions) generated by means of an iterative procedure for the answer of enormous linear structures of equations with a symmetric matrix. The matrix might be optimistic yes or indefinite.
The computation of zeros of polynomials is a classical challenge in numerical research. The contribution by way of Ammar, Calvetti, Gragg, and Reichel describes algorithms in accordance with Szegõ polynomials. particularly, wisdom of the site of zeros of Szegõ polynomials is necessary for the research and implementation of filters for time series.
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