Abstract
We study the relationship between sampling sequences in infinite dimensional Hilbert spaces of analytic functions and Marcinkiewicz–Zygmund inequalities in subspaces of polynomials. We focus on the study of the Hardy space and the Bergman space in one variable because they provide two settings with a strikingly different behavior.
Highlights
Marcinkiewicz–Zygmund inequalities are finite-dimensional models for sampling in an infinite dimensional Hilbert or Banach space of functions. They were studied in the context of interpolation by trigonometric polynomials
They became prominent in approximation theory, where they appear in quadrature rules and least square problems, and were usually studied in the context of orthogonal polynomials
The theorem shows that the construction of Marcinkiewicz–Zygmund families for the Bergman space is on the same level of difficulty as the construction of sampling sets for A2
Summary
Marcinkiewicz–Zygmund inequalities are finite-dimensional models for sampling in an infinite dimensional Hilbert or Banach space of functions They were studied in the context of interpolation by trigonometric polynomials. The natural finite-dimensional subspaces will be the family of the polynomials Pn of degree n In this context it is clear that an arbitrary set of at least n + 1 distinct points yields a sampling inequality. In our case H = A2(D) or = H 2(D) Both the Bergman space and the Hardy space are reproducing kernel Hilbert spaces, in which the polynomials are dense and kn(λ, λ) → k(λ, λ) pointwise. The theorem shows that the construction of Marcinkiewicz–Zygmund families for the Bergman space is on the same level of difficulty as the construction of sampling sets for A2.
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