Function spaces obeying a time-varying bandlimit

Abstract

Motivated by applications to signal processing and mathematical physics, recent work on the concept of time-varying bandwidth has produced a class of function spaces which generalize the Paley-Wiener spaces of bandlimited functions: any regular simple symmetric linear transformation with deficiency indices (1, 1) is naturally represented as multiplication by the independent variable in one of these spaces. We explicitly demonstrate the equivalence of this model for such linear transformations to several other functional models based on the theories of meromorphic model spaces of Hardy space and purely atomic Herglotz measures on the real line, respectively. This theory provides a precise notion of a time-varying or local bandwidth, and we describe how it may be applied to construct signal processing techniques that are adapted to signals obeying a time-varying bandlimit.

Jumping champions and prime gaps using information-theoretic tools

“…The generalized sampling theory [14][15][16][17][18][19][20][21] generalizes the regular Shannon sampling theorem of functions, or signals, that possess a constant bandwidth and constant Nyquist rate to classes of functions that possess a time-varying bandwidth, or, correspondingly, a timevarying Nyquist rate. This allows one to consider classes of signals of time-varying bandwidth that can be most stably reconstructed from their amplitudes on a sampling lattice, {t n }, whose spacing correspondingly varies in time.…”

“…The reconstruction formula Eq. (III.1) can now be applied with the generalized reconstruction kernel [14][15][16][17][18][19][20][21] :…”

“…is a smooth (infinitely differentiable) positive kernel function on R × R in the sense of reproducing kernel Hilbert space (RKHS) theory, and our space of functions obeying a 'generalized' bandlimit is the unique RKHS H(G) corresponding to G, see 20 (Section 2). The theory of these spaces is closely connected to the theory of Hardy spaces of analytic functions in the complex upper-half plane 20,[22][23][24][25][26][27][28][29][30][31] : Given any generalized RKHS, H(G), of time-varying or locally bandlimited functions, one can find a fixed function, M (t), so that multiplication by M (t) is a unitary transformation of H(G) onto a co-invariant (for the shift) subspace of the Hardy space of the upper half-plane which is the orthogonal complement of the range of a meromorphic inner function 20 . This inner function can be expressed explicitly in terms of the sequences of sample points {t n }, and their 'speeds' {t n } 20 .…”

“…The theory of these spaces is closely connected to the theory of Hardy spaces of analytic functions in the complex upper-half plane 20,[22][23][24][25][26][27][28][29][30][31] : Given any generalized RKHS, H(G), of time-varying or locally bandlimited functions, one can find a fixed function, M (t), so that multiplication by M (t) is a unitary transformation of H(G) onto a co-invariant (for the shift) subspace of the Hardy space of the upper half-plane which is the orthogonal complement of the range of a meromorphic inner function 20 . This inner function can be expressed explicitly in terms of the sequences of sample points {t n }, and their 'speeds' {t n } 20 .…”

Lorentz-covariant sampling theory for fields

“…The equivalence between bandlimitation and certain classes of GUPs for Minkowski spacetime has also been shown [101]. A general mathematical correspondence has been developed relating operators with finite minimal uncertainty and function spaces which exhibit a sampling theorem [20,21,[102][103][104]. This further motivated a notion of time-varying bandwidth, which has led to interesting applications in engineering [105][106][107][108] as well as connections to pure mathematics [109][110][111].…”