The continuity of superposition operators on some sequence spaces defined by moduli

Kolk, Enno; Raidjõe, Annemai

doi:10.1007/s10587-007-0075-3

Cited by 8 publications

(5 citation statements)

References 7 publications

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“…Proof. The statement on the range and the continuity follows from proposition 2.1 and theorem 3.1 in [28] with the modulus function = (t p ) k∈N . The injectivity of N p,q follows from lemma 3.2.…”

Section: P -Sparsity Regularizationmentioning

confidence: 96%

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On Tikhonov regularization with non-convex sparsity constraints

Zarzer

2009

Inverse Problems

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This paper deals with a theoretical analysis of a novel regularization technique for (nonlinear) inverse problems, in the field of the so-called sparsity promoting regularizations. We investigate the well-posedness and the convergence rates of a particular Tikhonov-type regularization. The regularization term is chosen to be the canonical norm in the sequence spaces p . In doing so we restrict ourselves to cases of 0 < p 1, motivated by sparsity promoting regularization. For p < 1 the triangle inequality is not valid any more and we are facing a non-convex constraint in a quasi Banach space. We provide results on the existence of minimizers, stability and convergence in a classic general setting. In addition we give convergence rates results in the respective Hilbert space topology under classic assumptions.

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Section: P -Sparsity Regularizationmentioning

confidence: 96%

“…The transformation of the minimization problem will be achieved via a particular nonlinear operator. The following definitions specify the operator as a so-called superposition operator (cf [27][28][29]), which eventually allows the desired transformation. Definition 3.1.…”

Section: P -Sparsity Regularizationmentioning

confidence: 99%

On Tikhonov regularization with non-convex sparsity constraints

Zarzer

2009

Inverse Problems

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“…Boundedness of the superposition operators on some sequence spaces was studied by Samae [16], Sagır and Güngör [14] and Chew [4], [5], [6], [8], [10], [11], [18]. Sagır and Güngör [15] characterized the superposition operators P g on C r0 (p) as follows …”

Section: If the Seriesmentioning

confidence: 99%

On characterization of boundedness of superposition operators on the Maddox space C_{r0}( p) of double sequences

Oğur¹

2017

ntmsci

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“…Dedagich and Zabreiko [7] have given the necessary and sufficient conditions for the superposition operators on the sequence spaces p , ∞ and c 0 . After, some properties of superposition operator, such as boundedness, compactness, were studied by Sama-ae [19], Sagır and Güngör [11,16], Kolk and Raidjoe [12], Ogur [14,15] and many others. The purpose of this paper is to generalize these works respect to the non-Newtonian calculus.…”

Section: Introductionmentioning

confidence: 99%

On characterization of non-Newtonian superposition operators in some sequence spaces

Sağır

Erdogan

2019

Filomat

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In this paper, we define a non-Newtonian superposition operator N P f where f : N × R(N) α → R(N) β by N P f (x) = f (k, x k) ∞ k=1 for every non-Newtonian real sequence x = (x k). Chew and Lee [4] have characterized P f : p → 1 and P f : c 0 → 1 for 1 ≤ p < ∞. The purpose of this paper is to generalize these works respect to the non-Newtonian calculus. We characterize N P f : ∞ (N) → 1 (N) , N P f : c 0 (N) → 1 (N) , N P f : c (N) → 1 (N) and N P f : p (N) → 1 (N), respectively. Then we show that such N P f : ∞ (N) → 1 (N) is *-continuous if and only if f (k, .) is *-continuous for every k ∈ N.

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The continuity of superposition operators on some sequence spaces defined by moduli

Cited by 8 publications

References 7 publications

On Tikhonov regularization with non-convex sparsity constraints

On Tikhonov regularization with non-convex sparsity constraints

On characterization of boundedness of superposition operators on the Maddox space C_{r0}( p) of double sequences

On characterization of non-Newtonian superposition operators in some sequence spaces

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