Convolution random sampling in multiply generated shift-invariant spaces of Lp(Rd)

被引:0
作者
Jiang, Yingchun [1 ]
Li, Wan [1 ]
机构
[1] Guilin Univ Elect Technol, Sch Math & Computat Sci, Guilin 541004, Peoples R China
基金
中国国家自然科学基金;
关键词
Multiply generated shift-invariant space; Convolution random sampling; Sampling stability; Condition number; Reconstruction algorithm;
D O I
10.1007/s43034-020-00098-2
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We mainly consider the stability and reconstruction of convolution random sampling in multiply generated shift-invariant subspaces V-p(Phi) = {Sigma(k is an element of Zd) c(k)(T) Phi(center dot -k) : (c(k))(k is an element of Zd)(l(p)(Z(d)))r} of L-p(R-d), 1 < p < infinity, where Phi = (phi(1), phi(2),., phi(r))(T) with phi(i) is an element of L-p(R-d) and c = (c(1), c(2),., c(r))(T) with c(i) is an element of l(p)(Z(d)), i = 1, 2,., r. The sampling set {x(j)}(j is an element of N) is randomly chosen with a general probability distribution over a bounded cube C-K and the samples are the form of convolution {f * psi (x j)}(j is an element of N) of the signal f. Under some proper conditions for the generator Phi, convolution function psi and probability density function rho, we first approximate V-p(Phi) by a finite dimensional subspace V-N(p)(Phi) ={Sigma(r)(i=1) Sigma(vertical bar k vertical bar <= N) c(i)(k)phi(i)(. - k) : c(i) is an element of l(p) ([-N, N](d))}. Then we show that the sampling stability holds with high probability for all functions in certain compact subsets V-K(p) (Phi) = {f is an element of V-p (Phi) : integral(CK) vertical bar f * psi(x)vertical bar(p) dx >= (1 - delta) integral(Rd) vertical bar f * psi(x)vertical bar (x)vertical bar p dx} of V-p(Phi) when the sampling size is large enough. Finally, we prove that the stability is related to the properties of the random matrix generated by {phi(i) * psi}1(<= i <= r) and give a reconstruction algorithm for the convolution random sampling of functions in V-N(p)(Phi).
引用
收藏
页数:22
相关论文
共 25 条
[11]   Compressed Sensing of Analog Signals in Shift-Invariant Spaces [J].
Eldar, Yonina C. .
IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2009, 57 (08) :2986-2997
[12]   Relevant sampling in finitely generated shift-invariant spaces [J].
Fuehr, Hartmut ;
Xian, Jun .
JOURNAL OF APPROXIMATION THEORY, 2019, 240 :1-15
[13]  
Hardle W., 1998, WAVELETS APPROXIMATI, V129, pxviii+265, DOI [10.1007/978-1-4612-2222-4, DOI 10.1007/978-1-4612-2222-4]
[14]   ON LINEAR INDEPENDENCE FOR INTEGER TRANSLATES OF A FINITE NUMBER OF FUNCTIONS [J].
JIA, RQ ;
MICCHELLI, CA .
PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, 1993, 36 :69-85
[15]   Stability of the shifts of a finite number of functions [J].
Jia, RQ .
JOURNAL OF APPROXIMATION THEORY, 1998, 95 (02) :194-202
[16]   Reconstruction from convolution random sampling in local shift invariant spaces [J].
Li, Yaxu ;
Wen, Jinming ;
Xian, Jun .
INVERSE PROBLEMS, 2019, 35 (12)
[17]   Non-uniform Random Sampling and Reconstruction in Signal Spaces with Finite Rate of Innovation [J].
Lu, Yancheng ;
Xian, Jun .
ACTA APPLICANDAE MATHEMATICAE, 2020, 169 (01) :247-277
[18]   Random sampling in reproducing kernel subspaces of Lp (Rn) [J].
Patel, Dhiraj ;
Sampath, Sivananthan .
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2020, 491 (01)
[19]   ONLINE LEARNING WITH MARKOV SAMPLING [J].
Smale, Steve ;
Zhou, Ding-Xuan .
ANALYSIS AND APPLICATIONS, 2009, 7 (01) :87-113
[20]   Reconstruction of functions in spline subspaces from local averages [J].
Sun, WC ;
Zhou, XW .
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2003, 131 (08) :2561-2571