Lions-Peetre reiteration formulas for triples and their applications

Irina Asekritova; Natan Krugljak; Lech Maligranda; Lyudmila Nikolova; Lars-Erik Persson

Displaying similar documents to “Lions-Peetre reiteration formulas for triples and their applications”

A Carlson type inequality with blocks and interpolation

Natan Kruglyak, Lech Maligranda, Lars Persson (1993)

Studia Mathematica

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An inequality, which generalizes and unifies some recently proved Carlson type inequalities, is proved. The inequality contains a certain number of “blocks” and it is shown that these blocks are, in a sense, optimal and cannot be removed or essentially changed. The proof is based on a special equivalent representation of a concave function (see [6, pp. 320-325]). Our Carlson type inequality is used to characterize Peetre’s interpolation functor $⟨ ⟩_{φ}$ (see [26]) and its Gagliardo closure...

A counterexample to the Γ-interpolation conjecture

Adama S. Kamara (2015)

Annales Polonici Mathematici

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Agler, Lykova and Young introduced a sequence $C_{ν}$ , where ν ≥ 0, of necessary conditions for the solvability of the finite interpolation problem for analytic functions from the open unit disc into the symmetrized bidisc Γ. They conjectured that condition $C_{n - 2}$ is necessary and sufficient for the solvability of an n-point interpolation problem. The aim of this article is to give a counterexample to that conjecture.

Interpolation methods of means and orbits

Mieczysław Mastyło (2005)

Studia Mathematica

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Banach operator ideal properties of the inclusion maps between Banach sequence spaces are used to study interpolation of orbit spaces. Relationships between those spaces and the method-of-means spaces generated by couples of weighted Banach sequence spaces with the weights determined by concave functions and their Janson sequences are shown. As an application we obtain the description of interpolation orbits in couples of weighted $L_{p}$ -spaces when they are not described by the K-method....

Interpolation of Cesàro sequence and function spaces

Sergey V. Astashkin, Lech Maligranda (2013)

Studia Mathematica

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The interpolation properties of Cesàro sequence and function spaces are investigated. It is shown that $C e s_{p} (I)$ is an interpolation space between $C e s_{p ₀} (I)$ and $C e s_{p ₁} (I)$ for 1 < p₀ < p₁ ≤ ∞ and 1/p = (1 - θ)/p₀ + θ/p₁ with 0 < θ < 1, where I = [0,∞) or [0,1]. The same result is true for Cesàro sequence spaces. On the other hand, $C e s_{p} [0, 1]$ is not an interpolation space between Ces₁[0,1] and $C e s_{\infty} [0, 1]$ .

Measure of weak noncompactness under complex interpolation

Andrzej Kryczka, Stanisław Prus (2001)

Studia Mathematica

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Logarithmic convexity of a measure of weak noncompactness for bounded linear operators under Calderón’s complex interpolation is proved. This is a quantitative version for weakly noncompact operators of the following: if T: A₀ → B₀ or T: A₁ → B₁ is weakly compact, then so is $T : A_{[θ]} \to B_{[θ]}$ for all 0 < θ < 1, where $A_{[θ]}$ and $B_{[θ]}$ are interpolation spaces with respect to the pairs (A₀,A₁) and (B₀,B₁). Some formulae for this measure and relations to other quantities measuring weak noncompactness are...

Sobolev-Besov spaces of measurable functions

Hans Triebel (2010)

Studia Mathematica

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The paper deals with spaces $L_{p}^{s} (ℝ ⁿ)$ of Sobolev type where s > 0, 0 < p ≤ ∞, and their relations to corresponding spaces $B_{p, q}^{s} (ℝ ⁿ)$ of Besov type where s > 0, 0 < p ≤ ∞, 0 < q ≤ ∞, in terms of embedding and real interpolation.

Polynomial interpolation and approximation in $ℂ^{d}$

T. Bloom, L. P. Bos, J.-P. Calvi, N. Levenberg (2012)

Annales Polonici Mathematici

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We update the state of the subject approximately 20 years after the publication of T. Bloom, L. Bos, C. Christensen, and N. Levenberg, Polynomial interpolation of holomorphic functions in ℂ and ℂⁿ, Rocky Mountain J. Math. 22 (1992), 441-470. This report is mostly a survey, with a sprinkling of assorted new results throughout.

Hausdorff dimension of a fractal interpolation function

Guantie Deng (2004)

Colloquium Mathematicae

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We obtain a lower bound for the Hausdorff dimension of the graph of a fractal interpolation function with interpolation points $(i / N, y_{i}) : i = 0, 1, . . ., N$ .

Complex interpolation of function spaces with general weights

Douadi Drihem (2023)

Commentationes Mathematicae Universitatis Carolinae

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We present the complex interpolation of Besov and Triebel–Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel–Lizorkin spaces. As a corollary of our results, we obtain the complex interpolation between the weighted Triebel–Lizorkin spaces ${\dot{F}}_{p_{0}, q_{0}}^{s_{0}} (ω_{0})$ and ${\dot{F}}_{\infty, q_{1}}^{s_{1}} (ω_{1})$ with suitable assumptions on the parameters $s_{0}, s_{1}, p_{0}, q_{0}$ and $q_{1}$ , and the pair of weights $(ω_{0}, ω_{1})$ .

On interpolation error on degenerating prismatic elements

Ali Khademi, Sergey Korotov, Jon Eivind Vatne (2018)

Applications of Mathematics

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We propose an analogue of the maximum angle condition (commonly used in finite element analysis for triangular and tetrahedral meshes) for the case of prismatic elements. Under this condition, prisms in the meshes may degenerate in certain ways, violating the so-called inscribed ball condition presented by P. G. Ciarlet (1978), but the interpolation error remains of the order $O (h)$ in the $H^{1}$ -norm for sufficiently smooth functions.

Interpolation of quasicontinuous functions

Joan Cerdà, Joaquim Martín, Pilar Silvestre (2011)

Banach Center Publications

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If C is a capacity on a measurable space, we prove that the restriction of the K-functional $K (t, f; L^{p} (C), L^{\infty} (C))$ to quasicontinuous functions f ∈ QC is equivalent to $K (t, f; L^{p} (C) \cap Q C, L^{\infty} (C) \cap Q C)$ . We apply this result to identify the interpolation space ${(L^{p ₀, q ₀} (C) \cap Q C, L^{p ₁, q ₁} (C) \cap Q C)}_{θ, q}$ .

A functional calculus description of real interpolation spaces for sectorial operators

Markus Haase (2005)

Studia Mathematica

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For a holomorphic function ψ defined on a sector we give a condition implying the identity ${(X, (A^{α}))}_{θ, p} = x \in X | t^{- θ R e α} ψ (t A) \in L ⁎^{p} ((0, \infty); X)$ where A is a sectorial operator on a Banach space X. This yields all common descriptions of the real interpolation spaces for sectorial operators and allows easy proofs of the moment inequalities and reiteration results for fractional powers.

$H^{\infty}$ functional calculus in real interpolation spaces, II

Giovanni Dore (2001)

Studia Mathematica

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Let A be a linear closed one-to-one operator in a complex Banach space X, having dense domain and dense range. If A is of type ω (i.e.the spectrum of A is contained in a sector of angle 2ω, symmetric about the real positive axis, and ${| | λ (λ I - A)}^{- 1} | |$ is bounded outside every larger sector), then A has a bounded $H^{\infty}$ functional calculus in the real interpolation spaces between X and the intersection of the domain and the range of the operator itself.

Several notes on the circumradius condition

Václav Kučera (2016)

Applications of Mathematics

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Recently, the so-called circumradius condition (or estimate) was derived, which is a new estimate of the $W^{1, p}$ -error of linear Lagrange interpolation on triangles in terms of their circumradius. The published proofs of the estimate are rather technical and do not allow clear, simple insight into the results. In this paper, we give a simple direct proof of the $p = \infty$ case. This allows us to make several observations such as on the optimality of the circumradius estimate. Furthermore, we show how...

$H^{\infty}$ functional calculus in real interpolation spaces

Giovanni Dore (1999)

Studia Mathematica

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Let A be a linear closed densely defined operator in a complex Banach space X. If A is of type ω (i.e. the spectrum of A is contained in a sector of angle 2ω, symmetric around the real positive axis, and $∥ λ {(λ I - A)}^{- 1} ∥$ is bounded outside every larger sector) and has a bounded inverse, then A has a bounded $H^{\infty}$ functional calculus in the real interpolation spaces between X and the domain of the operator itself.

Some results on function spaces of varying smoothness

Jan Schneider (2008)

Banach Center Publications

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This paper deals with function spaces of varying smoothness $B_{p}^{, s ₀} (ℝ ⁿ)$ , where the function :x ↦ s(x) determines the smoothness pointwise. Those spaces were defined in [2] and treated also in [3]. Here we prove results about interpolation, trace properties and present a characterization of these spaces based on differences.

Spherical basis function approximation with particular trend functions

Segeth, Karel

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The paper is concerned with the measurement of scalar physical quantities at nodes on the $(d - 1)$ -dimensional unit sphere surface in the $d$ -dimensional Euclidean space and the spherical RBF interpolation of the data obtained. In particular, we consider $d = 3$ . We employ an inverse multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 defined in Cartesian coordinates. We prove the existence of the interpolation formula of the type considered. The formula...

On BMO-regular couples of lattices of measurable functions

S. V. Kislyakov (2003)

Studia Mathematica

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We introduce a new “weak” BMO-regularity condition for couples (X,Y) of lattices of measurable functions on the circle (Definition 3, Section 9), describe it in terms of the lattice $X^{1 / 2} {(Y^{'})}^{1 / 2}$ , and prove that this condition still ensures “good” interpolation for the couple $(X_{A}, Y_{A})$ of the Hardy-type spaces corresponding to X and Y (Theorem 1, Section 9). Also, we present a neat version of Pisier’s approach to interpolation of Hardy-type subspaces (Theorem 2, Section 13). These two main results of the...

A recursive interpolation method in $R^{n}$

M. Ajzerle (1992)

Matematički Vesnik

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The Lizorkin-Freitag formula for several weighted $L_{p}$ spaces and vector-valued interpolation

Irina Asekritova, Natan Krugljak, Ludmila Nikolova (2005)

Studia Mathematica

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A complete description of the real interpolation space $L = {(L_{p ₀} (ω ₀), . . ., L_{p ₙ} (ω ₙ))}_{θ ⃗, q}$ is given. An interesting feature of the result is that the whole measure space (Ω,μ) can be divided into disjoint pieces $Ω_{i}$ (i ∈ I) such that L is an $l_{q}$ sum of the restrictions of L to $Ω_{i}$ , and L on each $Ω_{i}$ is a result of interpolation of just two weighted $L_{p}$ spaces. The proof is based on a generalization of some recent results of the first two authors concerning real interpolation of vector-valued spaces.

Interpolation and extension of Lipschitz-Hölder maps on $C_{p}$ spaces

Charles E. C Cleaver (1973)

Colloquium Mathematicae

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