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Sampling the flow of a bandlimited function
Sampling the flow of a bandlimited function
- Publication Year :
- 2020
- Publisher :
- HAL CCSD, 2020.
-
Abstract
- We analyze the problem of reconstruction of a bandlimited function $f$ from the space-time samples of its states $f_t=\phi_t\ast f$ 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-Tur\'an type inequalities. En route, we obtain a Remez-Tur\'an inequality for prolate spheroidal wave functions. To illustrate our results, we present some numerics and explicit estimates for the heat flow problem.<br />Comment: 29 pages
- Subjects :
- Signal Processing (eess.SP)
010103 numerical & computational mathematics
[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA]
Type (model theory)
[MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA]
01 natural sciences
Stability (probability)
Convolution
symbols.namesake
Bandlimited function
Classical Analysis and ODEs (math.CA)
FOS: Mathematics
FOS: Electrical engineering, electronic engineering, information engineering
Complex Variables (math.CV)
Electrical Engineering and Systems Science - Signal Processing
0101 mathematics
Condition number
Mathematics
Dynamical sampling
Mathematics - Complex Variables
Remez-Turan inequality
010102 general mathematics
Mathematical analysis
[MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV]
Function (mathematics)
Functional Analysis (math.FA)
Mathematics - Functional Analysis
Kernel (image processing)
Flow (mathematics)
Mathematics - Classical Analysis and ODEs
Fourier analysis
symbols
Geometry and Topology
[SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing
Mobile sampling
Heat flow
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....4941fbb2f4f4c29a25b13e3d4c33516a