Abstract
We analyze the problem of reconstruction of a bandlimited function f from the space–time samples of its states f_t=phi _t*f resulting from the convolution with a kernel phi _t. It is well-known that, in natural phenomena, uniform space–time samples of f are not sufficient to reconstruct f in a stable way. To enable stable reconstruction, a space–time sampling with periodic nonuniformly spaced samples must be used as was shown by Lu and Vetterli. We show that the stability of reconstruction, as measured by a condition number, controls the maximal gap between the spacial samples. We provide a quantitative statement of this result. In addition, instead of irregular space–time samples, we show that uniform dynamical samples at sub-Nyquist spatial rate allow one to stably reconstruct the function widehat{f} away from certain, explicitly described blind spots. We also consider several classes of finite dimensional subsets of bandlimited functions in which the stable reconstruction is possible, even inside the blind spots. We obtain quantitative estimates for it using Remez-Turán type inequalities. En route, we obtain Remez-Turán inequality for prolate spheroidal wave functions. To illustrate our results, we present some numerics and explicit estimates for the heat flow problem.
Highlights
We consider the sampling and reconstruction problem of signals u = u(t, x) that arise as an evolution of an initial signal f = f (x) under the action of convolution operators
The functions u are solutions of initial value problems stemming from a physical system
An analysis along the lines of the Papoulis sampling theorem [18] shows that the diffusion samples (1.6) of a function f ∈ P Wc do not lead to a stable recovery of f. These samples do allow for the stable recovery away from certain blind spots determined by φ; that is, one can effectively recover f · 1E, for a certain subset E ⊆ I of positive measure (1E denotes the characteristic function of the set E)
Summary
We consider the sampling and reconstruction problem of signals u = u(t, x) that arise as an evolution of an initial signal f = f (x) under the action of convolution operators. The functions u are solutions of initial value problems stemming from a physical system. The stable recovery of f from (1.3) amounts to finding conditions on , φ and L such that T has a bounded inverse from T (P Wc) to P Wc or, equivalently, the existence of A, B > 0 such that. |( f ∗ φt )(λ)|2 dt ≤ B f 22, for all f ∈ P Wc. If for a given φ and L the frame condition (1.4) is satisfied, we say that = φ,L is a stable sampling set. Let us discuss Problem 1 in more detail in the case of our prototypical example
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