Interpolation of nonlinear operators in weighted L p -spaces

Kaplitskii, Vitalii Markovich

doi:10.1007/s11202-010-0025-4

Cited by 2 publications

(3 citation statements)

References 5 publications

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“…It is shown in [4] that if x is a finitary vector, then the obtained estimate for K-functionals implies the required interpolation inequality for the operator T . Hence, as was done above, we pass to an estimate on the cone that approximates Q ∩ E + and consists of finitary vectors.…”

Section: Then the Family Of Triples Of Conesmentioning

confidence: 91%

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To the Theory of Operators that are Bounded on Cones in Weighted Spaces of Numerical Sequences

Kaplitskii

Dronov

2015

J Math Sci

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The paper is devoted to the general problem of obtaining interpolation theorems for operators that are bounded on cones in normed spaces and to some specific results pertaining to the particular problem of interpolation of operators that are bounded on cones in weighted spaces of numerical sequences. This setting is a natural generalization of the classical problem of interpolation of the boundedness property for a linear operator that is bounded between two Banach couples. We introduce the general concept of a Banach triple of cones possessing the interpolation property with respect to a given Banach triple. We provide sufficient conditions under which a triple of cones (Q 0 , Q 1 , Q) in weighted spaces of numerical sequences possesses the interpolation property with respect to a given Banach triple of weighted spaces of numerical sequences (F 0 , F 1 , F ). Appropriate interpolation theorems generalize the classical result about interpolation of linear operators in weighted spaces and are of interest for the theory of bases in Fréchet spaces. Bibliography: 10 titles.

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Section: Then the Family Of Triples Of Conesmentioning

confidence: 91%

“…It is shown in [4] that the above estimate for K-functionals implies the following interpolation inequality:…”

Section: Then the Family Of Triples Of Conesmentioning

confidence: 95%

To the Theory of Operators that are Bounded on Cones in Weighted Spaces of Numerical Sequences

Kaplitskii

Dronov

2015

J Math Sci

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“…Differently from interpolation methods in spaces of finite dimension (e.g., d-dimensional Euclidean spaces), interpolation here is in a space of functions, i.e., the interpolation nodes θ k (x) are functions in a Hilbert or a Banach space. Over the years, the problem of constructing a functional interpolant through suitable nodes in Hilbert or Banach spaces has been studied by several authors and convergence results were established in rather general cases [134,103,183,178,99,4,106,104,179,216]. Before discussing functional interpolation in detail, let us provide some geometric intuition on what functional interpolation is and what kind of representations we should expect.…”

Section: Functional Interpolation Methodsmentioning

confidence: 99%

The numerical approximation of nonlinear functionals and functional differential equations

Venturi

2018

Physics Reports

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We develop a rigorous convergence analysis for finite-dimensional approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs) defined on compact subsets of real separable Hilbert spaces. The purpose this analysis is twofold: first, we prove that under rather mild conditions, nonlinear functionals, functional derivatives and FDEs can be approximated uniformly by high-dimensional multivariate functions and high-dimensional partial differential equations (PDEs), respectively. Second, we prove that such functional approximations can converge exponentially fast, depending on the regularity of the functional (in particular its Fréchet differentiability), and its domain. We also provide sufficient conditions for consistency, stability and convergence of functional approximation schemes to compute the solution of FDEs, thus extending the Lax-Richtmyer theorem from PDEs to FDEs. Numerical applications are presented and discussed for prototype nonlinear functional approximation problems, and for linear FDEs.

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Interpolation of nonlinear operators in weighted L p -spaces

Cited by 2 publications

References 5 publications

To the Theory of Operators that are Bounded on Cones in Weighted Spaces of Numerical Sequences

To the Theory of Operators that are Bounded on Cones in Weighted Spaces of Numerical Sequences

The numerical approximation of nonlinear functionals and functional differential equations

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