Integral norm discretization and related problems
For researchers in approximation theory and numerical analysis, the paper provides both new findings and a comprehensive overview of norm discretization problems.
The paper studies discretization of integral and uniform norms for finite-dimensional spaces, focusing on multivariate trigonometric polynomials. It presents new results alongside a survey of known results.
The problem of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure is discussed in the paper. The above problem is studied for elements of finite dimensional spaces. Also, discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of the multivariate trigonometric polynomials with frequencies from a finite set with fixed cardinality. Both new results and a survey of known results are presented.