Gauss Quadrature for Freud Weights, Modulation Spaces, and Marcinkiewicz-Zygmund Inequalities

@article{Ehler2022GaussQF,
  title={Gauss Quadrature for Freud Weights, Modulation Spaces, and Marcinkiewicz-Zygmund Inequalities},
  author={Martin Ehler and Karlheinz Gr{\"o}chenig},
  journal={ArXiv},
  year={2022},
  volume={abs/2208.01122}
}
. We study Gauss quadrature for Freud weights and derive worst case error estimates for functions in a family of associated Sobolev spaces. For the Gaussian weight e − πx 2 these spaces coincide with a class of modulation spaces which are well-known in (time-frequency) analysis and also appear under the name of Hermite spaces. Extensions are given to more general sets of nodes that are derived from Marcinkiewicz-Zygmund inequalities. This generalization can be interpreted as a stability result… 

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References

SHOWING 1-10 OF 48 REFERENCES

Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature

Here, quadrature formulas are obtained that are exact for spherical harmonics of a fixed order, have nonnegative weights, and are based on function values at scattered sites for the unit sphere embedded in R q.

Sampling, Marcinkiewicz–Zygmund inequalities, approximation, and quadrature rules

On the Optimal Order of Integration in Hermite Spaces with Finite Smoothness

Higher order digital nets from the unit cube to a suitable subcube of $\mathbb{R}^s$ via a linear transformation are mapped and it is shown that such rules achieve, apart from powers of $\log N$, the optimal rate of convergence of the integration error.

Sub-optimality of Gauss-Hermite quadrature and optimality of trapezoidal rule for functions with finite smoothness

A sub-optimality of Gauss–Hermite quadrature and an optimality of the trapezoidal rule are proved in the weighted Sobolev spaces of square integrable functions of order α, where the optimality is in

Christoffel functions, orthogonal polynomials, and Nevai's conjecture for Freud weights

AbstractWe obtain upper and lower bounds for Christoffel functions for Freud weights by relatively new methods, including a new way to estimate discretization of potentials. We then deduce bounds for

Lectures on Hermite and Laguerre expansions

The interplay between analysis on Lie groups and the theory of special functions is well known. This monograph deals with the case of the Heisenberg group and the related expansions in terms of

Bounds for orthogonal polynomials for exponential weights

A note on Gauss—Hermite quadrature

SUMMARY For Gauss-Hermite quadrature, we consider a systematic method for transforming the variable of integration so that the integrand is sampled in an appropriate region. The effectiveness of the

Polynomial interpolation and hyperinterpolation over general regions

  • I. Sloan
  • Mathematics, Computer Science
  • 1995
Abstract This paper studies a generalization of polynomial interpolation: given a continuous function over a rather general manifold, hyperinterpolation is a linear approximation that makes use of