Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants

02/22/2016 - 13:30
02/22/2016 - 14:30
Speaker: 
Catherine Beneteau, University of South Florida
Location: 
McGill, Pav. Burnside, 805 O., rue Sherbrooke, salle 920
Abstract: 

 In this talk, I will discuss some polynomials that solve a particular optimization problem in Dirichlet-type spaces. Examples of Dirichlet-type spaces include the Hardy space (analytic functions in the disk whose coefficients are square summable), the Bergman space (analytic functions whose modulus squared is integrable with respect to area measure over the whole disk), and the (classical) Dirichlet space (analytic functions in the disk whose image has finite area, counting multiplicity). The optimal approximants p(z) in question minimize the Dirichlet- type norm of p(z) f(z) - 1, for a given function f(z). I will examine the connections between these optimal approximants, orthogonal polynomials and reproducing kernels, and exploit these connections to describe what is currently known about the zeros.

This work is joint with D. Khavinson, C. Liaw, D. Seco, and A. Sola. In this talk, I will discuss some polynomials that solve a particular optimization problem in Dirichlet-type spaces. Examples of Dirichlet-type spaces include the Hardy space (analytic functions in the disk whose coefficients are square summable), the Bergman space (analytic functions whose modulus squared is integrable with respect to area measure over the whole disk), and the (classical) Dirichlet space (analytic functions in the disk whose image has finite area, counting multiplicity). The optimal approximants p(z) in question minimize the Dirichlet- type norm of p(z) f(z) - 1, for a given function f(z). I will examine the connections between these optimal approximants, orthogonal polynomials and reproducing kernels, and exploit these connections to describe what is currently known about the zeros.

This work is joint with D. Khavinson, C. Liaw, D. Seco, and A. Sola. In this talk, I will discuss some polynomials that solve a particular optimization problem in Dirichlet-type spaces. Examples of Dirichlet-type spaces include the Hardy space (analytic functions in the disk whose coefficients are square summable), the Bergman space (analytic functions whose modulus squared is integrable with respect to area measure over the whole disk), and the (classical) Dirichlet space (analytic functions in the disk whose image has finite area, counting multiplicity). The optimal approximants p(z) in question minimize the Dirichlet- type norm of p(z) f(z) - 1, for a given function f(z). I will examine the connections between these optimal approximants, orthogonal polynomials and reproducing kernels, and exploit these connections to describe what is currently known about the zeros.

This work is joint with D. Khavinson, C. Liaw, D. Seco, and A. Sola. 

Last edited by on Thu, 02/18/2016 - 14:30