The split variational inclusion problem (SVIP) has been extensively studied and applied in real-world problems such as intensity-modulated radiation therapy (IMRT) and in sensor networks and in computerized tomography and data compression. Inspired by the works of López et al.[24], Byrne et al.[10] and Sitthithakerngkiet et al.[34], as well as of Moudafi and Thukur[29], we propose a self-adaptive step size algorithm for solving split variational inclusion problem (SVIP) without the prior knowledge of the operator norms.Under more mild conditions we obtain weak convergence of the proposed algorithm. We also construct a self-adaptive step size two-step iterative algorithm which converges strongly to the minimum-norm element of the solution of the SVIP. Finally, the performances and computational examples are presented and a comparison with related algorithms is provided to illustrate the efficiency and applicability of our new algorithms.
Sampling and reconstruction of signals in a shift-invariant space are generally studied under the requirement that the generator is in a stronger Wiener amalgam space, and the error estimates are usually given in the sense of $L_{p,{1 / \omega }}$
L
p
,
1
/
ω
-norm. Since we often need to reflect the local characteristics of reconstructing error, the asymptotic pointwise error estimates for nonuniform and average sampling in a non-decaying shift-invariant space are discussed under the assumption that the generator is in a hybrid-norm space. Based on Lemma 2.1–Lemma 2.6, we first rewrite the iterative reconstruction algorithms for two kinds of average sampling functionals and prove their convergence. Then, the asymptotic pointwise error estimates are presented for two algorithms under the case that the average samples are corrupted by noise.
We introduce a unified general iterative method to approximate a fixed point ofk-strictly pseudononspreading mapping. Under some suitable conditions, we prove that the iterative sequence generated by the proposed method converges strongly to a fixed point of ak-strictly pseudononspreading mapping with an idea of mean convergence, which also solves a class of variational inequalities as an optimality condition for a minimization problem. The results presented in this paper may be viewed as a refinement and as important generalizations of the previously known results announced by many other authors.
This paper mainly studies the average sampling and reconstruction in
shift-invariant subspaces of mixed Lebesgue spaces
$L^{p,q}(\mathbb{R}^{d+1})$, under the
condition that the generator $\varphi$ of the
shift-invariant subspace belongs to a hybrid-norm space of mixed form,
which is weaker than the usual assumption of Wiener amalgam space and
allows to control the orders $p,q$. First, the sampling stability for
two kinds of average sampling functionals are established. Then, we give
the corresponding iterative approximation projection algorithms with
exponential convergence for recovering the time-varying shift-invariant
signals from the average samples.
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