In this paper we consider polynomials orthogonal with respect to the linear functional L:P→C,defined on the space of all algebraic polynomials P by L[p]=∫_(-1)^(1)p(x)(1−x)^(α−1/2)(1+x)^(β−1/2)exp(iζx)dx,whereα,...In this paper we consider polynomials orthogonal with respect to the linear functional L:P→C,defined on the space of all algebraic polynomials P by L[p]=∫_(-1)^(1)p(x)(1−x)^(α−1/2)(1+x)^(β−1/2)exp(iζx)dx,whereα,β>−1/2 are real numbers such thatℓ=|β−α|is a positive integer,andζ∈R\{0}.We prove the existence of such orthogonal polynomials for some pairs ofαandζand for all nonnegative integersℓ.For such orthogonal polynomials we derive three-term recurrence relations and also some differential-difference relations.For such orthogonal polynomials the corresponding quadrature rules of Gaussian type are considered.Also,some numerical examples are included.展开更多
Abstract Asymptotic estimations of the Christoffel type functions for Lm extremal polynomials with an even integer m associated with generalized Jacobi weights are established. Also, asymptotic behavior of the zeros o...Abstract Asymptotic estimations of the Christoffel type functions for Lm extremal polynomials with an even integer m associated with generalized Jacobi weights are established. Also, asymptotic behavior of the zeros of the Lm extremal polynomials and the Cotes numbers of the corresponding Turán quadrature formula is given.展开更多
In this paper, we have proved some special properties of singular integral operators which are transformed from the singular integral equation defined in the interval (?1, 1), i.e., the properties of singular intergra...In this paper, we have proved some special properties of singular integral operators which are transformed from the singular integral equation defined in the interval (?1, 1), i.e., the properties of singular intergral operators at the endpoints and in the inner of (?1, 1).展开更多
基金supported in part by Serbian Ministry of Education and Science(Projects#174015 and Ⅲ44006).
文摘In this paper we consider polynomials orthogonal with respect to the linear functional L:P→C,defined on the space of all algebraic polynomials P by L[p]=∫_(-1)^(1)p(x)(1−x)^(α−1/2)(1+x)^(β−1/2)exp(iζx)dx,whereα,β>−1/2 are real numbers such thatℓ=|β−α|is a positive integer,andζ∈R\{0}.We prove the existence of such orthogonal polynomials for some pairs ofαandζand for all nonnegative integersℓ.For such orthogonal polynomials we derive three-term recurrence relations and also some differential-difference relations.For such orthogonal polynomials the corresponding quadrature rules of Gaussian type are considered.Also,some numerical examples are included.
基金Supported by the National Natural Science Foundation of China (No.19971089).
文摘Abstract Asymptotic estimations of the Christoffel type functions for Lm extremal polynomials with an even integer m associated with generalized Jacobi weights are established. Also, asymptotic behavior of the zeros of the Lm extremal polynomials and the Cotes numbers of the corresponding Turán quadrature formula is given.
文摘In this paper, we have proved some special properties of singular integral operators which are transformed from the singular integral equation defined in the interval (?1, 1), i.e., the properties of singular intergral operators at the endpoints and in the inner of (?1, 1).
