Non-standard anti-Gaussian quadrature rules

[N. Z. Petrović]

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Ova disertacija se bavi generalizacijom anti-Gauss-ovih kvadraturnih formula koje su uvedene 1996. godine na prostoru algebarskih polinoma. Prva generalizacija koja je urađena odnosi se na proširenje ovih formula na prostor trigonometrijskih polinoma, pri čemu je posebna pažnja posvećena parnim težinskim funkcijama na intervalu [−π, π). Glavne osobine ovih kvadraturnih formula su dokazane i predstavljen je efikasan numerički metod za njihovo konstruisanje. Taj metod je baziran na vezama između čvorova i težinskih koeficijenata kvadraturne formule za trigonometrijske polinome i odgovarajuće kvadraturne formule za algebarske polinome. Druga vrsta generalizacije koja je predstavljena u ovoj disertaciji odnosi se na višestruku ortogonalnost. Naime, uvedeni su pojmovi skupa anti-Gauss-ovih i skupa usrednjenih kvadraturnih formula za optimalni skup kvadraturnih formula u Borges-ovom smislu, kao i odgovarajuća klasa višestruko ortogonalnih polinoma...

In this thesis we are considering the generalization of anti-Gaussian quadrature rules, introduced for algebraic polynomials in 1996. The first generalization refers to the extension of these formulas to the space of trigonometric polynomials, with a special attention paid to an even weight function on [−π, π). The main properties of such quadrature rules have been proved, and an effective numerical method for their construction has been presented. That method is based on relations between nodes and weights of the quadrature rule for trigonometric polynomials, and the corresponding quadrature rule for algebraic polynomials. The second type of generalization presented in this dissertation is related to multiple orthogonality. Namely, the notions of a set of anti-Gaussian and a set of averaged quadrature rules for the optimal set of quadrature rules in Borges’ sense were introduced, as well as the corresponding class of multiple orthogonal polynomials. The main properties of such quadrature rules and multiple orthogonal polynomials have been proved, and numerical methods for their constructions have been presented. Both generalizations include some numerical examples, and the extension to the space of trigonometric polynomials is completed by a comparison with other available methods.

anti-Gaussian quadrature rules, recurrence relation, weight functions, averaged Gaussian quadrature rule, optimal set of quadrature rules in Borges’ sense, multiple orthogonal polynomials

519.6(043.3)

Matematika - Numerička analiza

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