Related papers: Universal truncation error upper bounds in samplin…
Universal (pointwise uniform and time shifted) truncation error upper bounds are presented in Whittaker--Kotel'nikov--Shannon (WKS) sampling restoration sum for Bernstein function class $B_{\pi,d}^q\,,\ q \ge 1,$ $d\in \mathbb N\,,$ when…
The article starts with generalizations of some classical results and new truncation error upper bounds in the sampling theorem for bandlimited stochastic processes. Then, it investigates $L_p([0,T])$ and uniform approximations of…
The harmonizable Piranashvili-type stochastic processes are approximated by finite time shifted average sampling sums. Explicit truncation error upper bounds are established. Various corollaries and special cases are discussed.
The Shannon sampling theorem for bandlimited wide sense stationary random processes was established in 1957, which and its extensions to various random processes have been widely studied since then. However, truncation of the Shannon series…
The article starts with new aliasing-truncation error upper bounds in the sampling theorem for non-bandlimited stochastic signals. Then, it investigates $L_p([0,T])$ approximations of sub-Gaussian random signals. Explicit truncation error…
Recently efforts have been made to use generalized sinc functions to perfectly reconstruct various kinds of non-bandlimited signals. As a consequence, perfect reconstruction sampling formulas have been established using such generalized…
The fast reconstruction of a bandlimited function from its sample data is an essential problem in signal processing. In this paper, we consider the widely used Gaussian regularized Shannon sampling formula in comparison to regularized…
Reconstructing a band-limited function from its finite sample data is a fundamental task in signal analysis. A Gaussian regularized Shannon sampling series has been proved to be able to achieve exponential convergence for uniform sampling.…
We consider the reconstruction of a bandlimited function from its finite localized sample data. Truncating the classical Shannon sampling series results in an unsatisfactory convergence rate due to the slow decay of the sinc function. To…
The classical sampling Nyquist-Shannon-Kotelnikov theorem states that a band-limited continuous time function can be uniquely recovered without error from a infinite two-sided sampling series taken with a sufficient frequency. This short…
We consider the problem of reconstructing a wide sense stationary band-limited process from its local averages taken either at the Nyquist rate or above. As a result, we obtain a sufficient condition under which average sampling expansions…
Weighted average sampling is more practical and numerically more stable than sampling at single points as in the classical Shannon sampling framework. Using the frame theory, one can completely reconstruct a bandlimited function from its…
We present a strategy for estimating the error of truncated functional flow equations. While the basic functional renormalization group equation is exact, approximated solutions by means of truncations do not only depend on the choice of…
To facilitate the numerical analysis of particle methods, we derive truncation error estimates for the approximate operators in a generalized particle method. Here, a generalized particle method is defined as a meshfree numerical method…
Uniform deviation bounds limit the difference between a model's expected loss and its loss on an empirical sample uniformly for all models in a learning problem. As such, they are a critical component to empirical risk minimization. In this…
For a class $F$ of complex-valued functions on a set $D$, we denote by $g_n(F)$ its sampling numbers, i.e., the minimal worst-case error on $F$, measured in $L_2$, that can be achieved with a recovery algorithm based on $n$ function…
We study some problems related to the effect of bounded, additive sample noise in the bandlimited interpolation given by the Whittaker-Shannon-Kotelnikov (WSK) sampling formula. We establish a generalized form of the WSK series that allows…
Assume that samples of a filtered version of a function in a shift-invariant space are avalaible. This work deals with the existence of a sampling formula involving these samples and having reconstruction functions with compact support.…
Recent findings by Jahn, T. Ullrich, Voigtlaender [10] relate non-linear sampling numbers for the square norm to quantities involving trigonometric best $m-$term approximation errors in the uniform norm. Here we establish new results for…
In this paper, we discuss some numerical realizations of Shannon's sampling theorem. First we show the poor convergence of classical Shannon sampling sums by presenting sharp upper and lower bounds of the norm of the Shannon sampling…