Uma classe de polinômios ortogonais de Sobolev no círculo unitário: limitantes, resultados assintóticos e zeros
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2022-02-16
Autores
Silva, Jéssica Ventura da
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Universidade Estadual Paulista (Unesp)
Resumo
O principal objetivo deste trabalho é apresentar um estudo detalhado de uma classe de polinômios ortogonais de Sobolev no círculo unitário introduzida por A. Sri Ranga em “Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle, Proc. Amer. Math. Soc., 144 (2016), 1129-1143”. A classe paramétrica de produtos internos de Sobolev em nosso estudo envolve pares de medidas conhecidas como “pares coerentes de medidas de segundo tipo no círculo unitário”. De acordo com este conceito de coerência, os polinômios ortogonais de Sobolev associados estão relacionados aos polinômios ortogonais com respeito a uma das medidas no produto interno de Sobolev por uma fórmula de conexão simples. Na presente tese consideramos outras propriedades, tais como limitantes e assintóticas, relacionadas aos coeficientes na fórmula de conexão e também aos polinômios ortogonais de Sobolev associados. Assumindo certas restrições nos parâmetros que definem o produto interno de Sobolev, mostramos também que os coeficientes de conexão estão relacionados a uma subclasse dos polinômios “continuous dual Hahn”. Além disso, determinamos algumas condições para que os zeros dos polinômios ortogonais de Sobolev e/ou de suas derivadas estejam dentro do disco unitário. Finalmente, usando experimentos numéricos, encontramos mais informações sobre os parâmetros para que os zeros de tais polinômios estejam dentro do disco unitário.
The main goal of this work is to present a detailed study of a class of Sobolev orthogonal polynomials on the unit circle introduced by A. Sri Ranga in “Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle, Proc. Amer. Math. Soc., 144 (2016), 1129-1143”. The parametric class of Sobolev inner products in our study involve pairs of measures known as “coherent pairs of measures of the second kind on the unit circle”. Under this concept of coherence, the associated Sobolev orthogonal polynomials are found to be related to the orthogonal polynomials associated with one of the measures in the Sobolev inner product by a simple connection formula. In the present thesis we consider further properties, such as bounds and asymptotics, regarding the coefficients in the connection formula and also the associated Sobolev orthogonal polynomials. Assuming certain restrictions on the parameters that define the Sobolev inner product, we also show that the connection coefficients are related to a subclass of the “continuous dual Hahn” polynomials. Furthermore, we determine some conditions so that the zeros of the Sobolev orthogonal polynomials and/or its derivatives stay within the unit disk. Finally, using numerical experiments, we find further information on the parameters so that the zeros of such polynomials are within the unit disk.
The main goal of this work is to present a detailed study of a class of Sobolev orthogonal polynomials on the unit circle introduced by A. Sri Ranga in “Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle, Proc. Amer. Math. Soc., 144 (2016), 1129-1143”. The parametric class of Sobolev inner products in our study involve pairs of measures known as “coherent pairs of measures of the second kind on the unit circle”. Under this concept of coherence, the associated Sobolev orthogonal polynomials are found to be related to the orthogonal polynomials associated with one of the measures in the Sobolev inner product by a simple connection formula. In the present thesis we consider further properties, such as bounds and asymptotics, regarding the coefficients in the connection formula and also the associated Sobolev orthogonal polynomials. Assuming certain restrictions on the parameters that define the Sobolev inner product, we also show that the connection coefficients are related to a subclass of the “continuous dual Hahn” polynomials. Furthermore, we determine some conditions so that the zeros of the Sobolev orthogonal polynomials and/or its derivatives stay within the unit disk. Finally, using numerical experiments, we find further information on the parameters so that the zeros of such polynomials are within the unit disk.
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Polinômios ortogonais de Sobolev no círculo unitário, Coeficientes de conexão, Polinômios "continuous dual Hahn", Sobolev orthogonal polynomials on the unit circle, Connection coefficients, Continuous dual Hahn polynomials