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requires the same level control over all the variables of a function in time domains or spatial fields 10 . Thus, for many time‐varying signals that depend on independent quantities with different properties in the practice, the mixed Lebesgue space

L^{p, q} (ℝ^{d + 1}) = \{f (x, y) : ‖ f ‖_{L^{p, q}} = {‖‖ f ‖_{L_{y}^{q} (ℝ^{d})}‖}_{L_{x}^{p} (ℝ)} < \infty\}

seems to be more suitable to model and measure those signals, due to its separate integrability for different variables.…”

Section: Introductionmentioning

confidence: 99%

“…The mixed Lebesgue spaces were first introduced in Benedek and Panzone 11 and generally studied in harmonic analysis and operator theory 12–16 . Recently, many sampling results for signals in the bandlimited, shift‐invariant, and reproducing kernel subspaces of the mixed Lebesgue space are obtained, which generalize the existing sampling results for signals in the corresponding subspaces of Lebesgue space 10,17–23 . However, the basis for obtaining those sampling results is that the L p , q norms of the signals are bounded, which prevents us from applying such techniques to signals that do not decay or even grow at infinity 24 .…”

Section: Introductionmentioning

confidence: 99%

Average sampling and reconstruction for signals in shift‐invariant subspaces of weighted mixed Lebesgue spaces

Wang

Zhang

2021

Math Methods in App Sciences

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In this paper, we mainly study the average sampling problem for signals in a shift‐invariant subspace of weighted mixed Lebesgue space. First, the sampling stability for two kinds of average sampling functionals is established. Then, two iterative reconstruction algorithms with exponential convergence are presented for recovering the shift‐invariant signals from the corresponding average samples.

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“…In fact, Sampling of band-limited signals in mixed Lebesgue spaces was studied in [22,24]. Recently, nonuniform sampling in shift-invariant subspaces of L p,q (R d+1 ) was discussed in [16].In this paper, we study the sampling and reconstruction of signals in a reproducing kernel subspace of L p,q (R d+1 ). The classical sampling sets are not adaptive to signals and the sampling process is linear.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

Adaptive sampling of time-space signals in a reproducing kernel subspace of mixed Lebesgue space

Jiang

Sun

2020

Banach J. Math. Anal.

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The Mixed Lebesgue space is a suitable tool for modelling and measuring signals living in time-space domains. And sampling in such spaces plays an important role for processing high-dimensional time-varying signals. In this paper, we first define reproducing kernel subspaces of mixed Lebesgue spaces. Then, we study the frame properties and show that the reproducing kernel subspace has finite rate of innovation. Finally, we propose a semi-adaptive sampling scheme for time-space signals in a reproducing kernel subspace, where the sampling in time domain is conducted by a time encoding machine. Two kinds of timing sampling methods are considered and the corresponding iterative approximation algorithms with exponential convergence are given.Sampling is an important task in signal and image processing. There are many results for sampling and reconstruction of various signals, such as bandlimited signals [19,21], signals in shift-invariant spaces [1,17,23], signals with finite rate of innovation [20] and signals in a reproducing kernel subspace [10,11,18,25]. Sampling for signals living in a mixed Lebesgue space is useful for processing time-based signals. In fact, Sampling of band-limited signals in mixed Lebesgue spaces was studied in [22,24]. Recently, nonuniform sampling in shift-invariant subspaces of L p,q (R d+1 ) was discussed in [16].In this paper, we study the sampling and reconstruction of signals in a reproducing kernel subspace of L p,q (R d+1 ). The classical sampling sets are not adaptive to signals and the sampling process is linear. Recently, a time sampling approach called time encoding machine (TEM) has received attention [12,13,14,15], which is inspired by the neurons models. Instead of recording the value of a signal f (t) at a preset time instant, one records the time at which the signal takes on a preset value. So it is a signal-dependent and nonlinear sampling mechanism. It is more practical in practice, due to its simplicity and low-cost for sampling. A time encoding machine maps amplitude information of a signal into the timing domain, which was first introduced by Lazar and Tóth in [12] for the special case of bandlimited signals and was extended to L 2 -shiftinvariant subspaces [9] and more general framework of weighted reproducing kernel subspaces [11].Since the signals f (x, y) in our setting live in the time-space domains, we assume that some sampling devices with function of time encoding are located at Γ = {y j : j ∈ J} ⊂ R d .Each device living on y j , j ∈ J first takes samples f (x, y j ) in space domain, and then produces samples for time domain by time encoding machines. Γ is supposed to be relatively-separated, that is,Here, J is a countable index set, B(y, δ ′ ) are ball in R d with radius δ ′ .This paper is organized as follows. In section 2, we define the reproducing kernel subspaces of mixed Lebesgue spaces L p,q (R d+1 ) and give a class of examples. In section 3, the frame properties of reproducing kernel subspaces are studied. Section 4 is devoted to presenting two kinds of...

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Nonuniform sampling in principal shift-invariant subspaces of mixed Lebesgue spaces Lp,q(Rd+1

Cited by 36 publications

References 18 publications

The closedness of shift invariant subspaces in L p , q ( R d + 1

The closedness of shift invariant subspaces in L p , q ( R d + 1

Average sampling and reconstruction for signals in shift‐invariant subspaces of weighted mixed Lebesgue spaces

Adaptive sampling of time-space signals in a reproducing kernel subspace of mixed Lebesgue space

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