Convolutional frames for sampling, signal recovery and uncertainty principles

Abstract

KEYWORDS: B-spline; Convolutional frame; Derivative sampling; Erasures; Fiberization map; Filter bank; Frame; Fusion frame; Locally compact group; Multi-channel sampling; Multiplication-invariant space; Periodic shiftinvariant space; Random sampling; Ramanujan filter bank; Ramanujan subspace; Ramanujan sums; Range function; Signal concentration; Supremum cosine angle; Tight frame; Translation-invariant space; Trigonometric polynomial; Twisted shift-invariant space; Uncertainty principle; Weyl-Zak transform; Zak transform. Sampling theory addresses the fundamental problem of determining whether a continuous function can be completely reconstructed from a discrete set of its values, commonly referred to as samples. The classical Shannon sampling theorem establishes that bandlimited functions are entirely determined by their values at integer points and can be reconstructed via sinc interpolation. Over the decades, this theory has been generalized to accommodate more realistic and flexible signal models, including nonuniform, derivative, multi-channel, and random sampling, as well as sampling in shift-invariant spaces. These developments have significantly broadened the scope of sampling theory, making it a unifying principle across communication, signal processing, medical imaging, geophysical sensing, machine learning, and quantum signal processing.