Convolution random sampling in multiply generated shift-invariant spaces of $$L^p(\mathbb {R}^{d})$$
- PDF / 1,898,723 Bytes
- 22 Pages / 439.37 x 666.142 pts Page_size
- 33 Downloads / 198 Views
Annals of Functional Analysis https://doi.org/10.1007/s43034-020-00098-2 ORIGINAL PAPER
Convolution random sampling in multiply generated shift‑invariant spaces of Lp (ℝd ) Yingchun Jiang1 · Wan Li1 Received: 23 May 2020 / Accepted: 16 October 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract We mainly consider the stability and reconstruction of convolution random sampling in multiply generated shift-invariant subspaces } { ∑ T p d r p c(k) 𝛷(⋅ − k) ∶ (c(k))k∈ℤd ∈ (𝓁 (ℤ )) V (𝛷) = k∈ℤd
of Lp (ℝd ) , 1 < p < ∞ , where 𝛷 = (𝜙1 , 𝜙2 , … , 𝜙r )T with 𝜙i ∈ Lp (ℝd ) and c = (c1 , c2 , … , cr )T with ci ∈ 𝓁 p (ℤd ) , i = 1, 2, … , r . The sampling set {xj }j∈ℕ is randomly chosen with a general probability distribution over a bounded cube CK and the samples are the form of convolution {f ∗ 𝜓(xj )}j∈ℕ of the signal f. Under some proper conditions for the generator 𝛷 , convolution function 𝜓 and probability density function 𝜌 , we first approximate V p (𝛷) by a finite dimensional subspace { r } ∑∑ p p d VN (𝛷) = ci (k)𝜙i (⋅ − k) ∶ ci ∈ 𝓁 ([−N, N] ) . i=1 |k|≤N
Then we show that the sampling stability holds with high probability for all functions in certain compact subsets
Communicated by Wenchang Sun. * Wan Li [email protected] Yingchun Jiang [email protected] 1
School of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004, People’s Republic of China Vol.:(0123456789)
Y. Jiang and W. Li
{ p
VK (𝛷) =
f ∈ V p (𝛷) ∶
�CK
|f ∗ 𝜓(x)|p dx ≥ (1 − 𝛿)
�ℝd
} |f ∗ 𝜓(x)|p dx
of V p (𝛷) when the sampling size is large enough. Finally, we prove that the stability is related to the properties of the random matrix generated by {𝜙i ∗ 𝜓}1≤i≤r and give a reconstruction algorithm for the convolution random sampling of functions in p VN (𝛷). Keywords Multiply generated shift-invariant space · Convolution random sampling · Sampling stability · Condition number · Reconstruction algorithm Mathematics Subject Classification 46E22 · 94A20
1 Introduction Random sampling plays an important role in many fields, such as image processing [9], compressed sensing [8, 11] and learning theory [19]. Nonuniform ideal random sampling has been generally studied for multivariate trigonometric polynomials [4], bandlimited signals [5, 6], signals that satisfy some locality properties in short-time Fourier transform [22], signals in a shift-invariant space [12, 23, 24], signals with finite rate of innovation in L2 (ℝd ) [17] and signals in a reproducing kernel subspace of Lp (ℝd ) [18]. Due to the physical characteristics of sampling devices, the available sampled values may not be the exact values of f on sampling set, while they are the local average of f. Although average sampling had been generally studied for all kinds of signals in the context of deterministic sampling, see [1, 2, 20, 21] and references therein, the special convolution average random sampling has only been considered in [16] for shift-invariant signals in L2 (ℝd ) . More precisely, the sampling set X = {xj ∶ j ∈
Data Loading...
