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SS 2025

29.04.2025 um 12:00 Uhr in 66/E01

Prof. Dr. Stefan Kunis (Universität Osnabrück)

Sign localized test functions, conditioning of Fourier matrices, and generalized quadrature

In this talk, we would like to discuss multivariate Fourier transform pairs where one function has bounded support and the transformed function has a well specified zero crossing. Such functions have widespread applications ranging from analytic number theory, sphere packing, geometry of quadrature points to computational diffraction limits in super-resolution microscopy.
Our first example is a smooth compactly supported bump function with Fourier transform being positive in a small ball and non-positive outside. Together with the Poisson summation formula, this allows to  estimate the smallest singular value of multivariate nonequispaced Fourier matrices when the points are separated with respect to the order of the Fourier matrix. We apply these to establish a relatively sharp computational diffraction limit.
The second example concerns a multivariate trigonometric polynomial, i.e. bounded support in frequency domain, being positive in a small ball and non-positive outside. For positive quadrature rules which are exact for trigonometric polynomials up to a certain degree, this allows to upper bound the covering radius of the quadrature nodes. We discuss generalizations to curve-length estimates for mobile sampling on the unit square, the torus and the sphere where curves of order-optimal length can be constructed from lattice-generating vectors. If time permits, we would like to discuss dispersion estimates of quadrature points for hyperbolic cross trigonometric polynomials.