Interpolation of Linear Operators

Крейн, С. Г.; Petunin, Ju.; Семенов, Э. М.

doi:10.1090/mmono/054

Cited by 615 publications

(1,009 citation statements)

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“…Observe that Ψ (and hencef ) depends on both p and w. Since C p,w is an order continuous space with the Fatou property its Köthe dual (C p,w ) ′ can be identified with its Banach dual space (C p,w ) * . In fact each bounded linear functional F ∈ (C p,w ) * is of the integral form F (g) = I f (t)g(t) dt, g ∈ C p,w , where f ∈ (C p,w ) ′ and F (Cp,w ) * = f (Cp,w) ′ [5,19]. We start with several preparatory lemmas.…”

Section: Description Of the Dual Spacementioning

confidence: 99%

On the dual of Cesàro function space

Kamińska

Kubiak

2012

Nonlinear Analysis: Theory, Methods & Applications

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Abstract. The goal of this paper is to present an isometric representation of the dual space to Cesàro function space Cp,w, 1 < p < ∞, induced by arbitrary positive weight function w on interval (0, l) where 0 < l ∞. For this purpose given a strictly decreasing nonnegative function Ψ on (0, l), the notion of essential Ψ-concave majorantf of a measurable function f is introduced and investigated. As applications it is shown that every slice of the unit ball of the Cesàro function space has diameter 2. Consequently Cesàro function spaces do not have the Radon-Nikodym property, are not locally uniformly convex and they are not dual spaces.

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Section: Description Of the Dual Spacementioning

confidence: 99%

On the dual of Cesàro function space

Kamińska

Kubiak

2012

Nonlinear Analysis: Theory, Methods & Applications

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“…For more properties of d x , x * and x * * see [1,23]. Let 0 < p < ∞ and w ∈ L 0 be a nonnegative weight function, the Lorentz space Γ p,w is a subspace of L 0 such that…”

Section: Preliminariesmentioning

confidence: 99%

“…The space Γ p,w is an interpolation space between L 1 and L ∞ yielded by the Lions-Peetre K-method [1,23]. Clearly, Γ p,w ⊂ Λ p,w .…”

Section: Preliminariesmentioning

confidence: 99%

“…Let 0 < λ < ∞ and φ be concave increasing function on [0, α) with φ(0 + ) = 0. Replacing x by f − c and y n by c n − c in Proposition 1.2 [23] we get the Lorentz space Λ φ with the property (S) equipped with the norm given by…”

Section: Property (S) and Continuity Of Metric Projection Operatormentioning

confidence: 99%

“…It is well-known that d * (w, 1) is the dual space of d(w, 1). For more details of the Lorentz and Marcinkiewicz spaces the reader is referred to [16,23,26 …”

Section: Properties Of Metric Projection Onto One-dimensional Subspacmentioning

confidence: 99%

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Best approximation properties in spaces of measurable functions

Ciesielski

Lewicki

2017

Mathematische Nachrichten

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Abstract. We research proximinality of µ-sequentially compact sets and µ-compact sets in measurable function spaces. Next we show a correspondence between the Kadec-Klee property for convergence in measure and µ-compactness of the sets in Banach function spaces. Also the property S is investigated in Fréchet spaces and employed to provide the Kadec-Klee property for local convergence in measure. We discuss complete criteria for continuity of metric projection in Fréchet spaces with respect to the Hausdorff distance. Finally, we present the necessary and sufficient condition for continuous metric selection onto a one-dimensional subspace in sequence Lorentz spaces d(w, 1). Mathematics Subjects Classification: 41A65, 46E30, 46A40.

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Convergence of Multigrid Algorithms for Interior Penalty Methods

Brenner

Zhao

2005

Appl. Num. Anal. Comp. Math.

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Key words multigrid algorithms, interior penalty methods, second order problems Subject classification 65N55, 65N30V -cycle, F -cycle and W -cycle multigrid algorithms for interior penalty methods for second order elliptic boundary value problems are studied in this paper. It is shown that these algorithms converge uniformly with respect to all grid levels if the number of smoothing steps is sufficiently large, and that the contraction numbers decrease as the number of smoothing steps increases, at a rate determined by the elliptic regularity of the problem.

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Interpolation of Linear Operators

Cited by 615 publications

References 0 publications

On the dual of Cesàro function space

On the dual of Cesàro function space

Best approximation properties in spaces of measurable functions

Convergence of Multigrid Algorithms for Interior Penalty Methods

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