Function spaces related to continuous negative definite functions: ψ-Bessel potential spaces

Walter Farkas; Niels Jacob; René L. Schilling

Displaying similar documents to “Function spaces related to continuous negative definite functions: ψ-Bessel potential spaces”

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.

Lions-Peetre reiteration formulas for triples and their applications

Irina Asekritova, Natan Krugljak, Lech Maligranda, Lyudmila Nikolova, Lars-Erik Persson (2001)

Studia Mathematica

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We present, discuss and apply two reiteration theorems for triples of quasi-Banach function lattices. Some interpolation results for block-Lorentz spaces and triples of weighted $L_{p}$ -spaces are proved. By using these results and a wavelet theory approach we calculate (θ,q)-spaces for triples of smooth function spaces (such as Besov spaces, Sobolev spaces, etc.). In contrast to the case of couples, for which even the scale of Besov spaces is not stable under interpolation, for triples we...

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...

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 $C e s_{\infty} [0, 1]$ .

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.

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 ${(L^{p ₀, q ₀} (C) \cap Q C, L^{p ₁, q ₁} (C) \cap Q C)}_{θ, q}$ .

$H^{\infty}$ 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.

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.

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.

$H^{\infty}$ 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.

Interpolation problems in cones. Nota I

Carlos A. Berenstein, Daniele Struppa (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In questa nota, si studiano problemi di interpolazione per varietà discrete in spazi di funzioni olomorfe in coni. In particolare si mostra come sia possibile estendere il Principio Fondamentale di Ehrenpreis ad equazioni di convoluzione nella spazio $H_{c}(\Omega)$ , introdotto in [4] in connessione con problemi di fisica quantistica.

IP-Dirichlet measures and IP-rigid dynamical systems: an approach via generalized Riesz products

Sophie Grivaux (2013)

Studia Mathematica

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If ${(n_{k})}_{k \geq 1}$ is a strictly increasing sequence of integers, a continuous probability measure σ on the unit circle is said to be IP-Dirichlet with respect to ${(n_{k})}_{k \geq 1}$ if $σ ̂ (\sum_{k \in F} n_{k}) \to 1$ as F runs over all non-empty finite subsets F of ℕ and the minimum of F tends to infinity. IP-Dirichlet measures and their connections with IP-rigid dynamical systems have recently been investigated by Aaronson, Hosseini and Lemańczyk. We simplify and generalize some of their results, using an approach involving generalized Riesz...

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 $(ω_{0}, ω_{1})$ .

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 $(i / N, y_{i}) : i = 0, 1, . . ., N$ .

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...

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

M. Ajzerle (1992)

Matematički Vesnik

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Anisotropic $h p$ -adaptive method based on interpolation error estimates in the $H^{1}$ -seminorm

Vít Dolejší (2015)

Applications of Mathematics

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We develop a new technique which, for the given smooth function, generates the anisotropic triangular grid and the corresponding polynomial approximation degrees based on the minimization of the interpolation error in the broken $H^{1}$ -seminorm. This technique can be employed for the numerical solution of boundary value problems with the aid of finite element methods. We present the theoretical background of this approach and show several numerical examples demonstrating the efficiency of...

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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On Synge-type angle condition for $d$ -simplices

Antti Hannukainen, Sergey Korotov, Michal Křížek (2017)

Applications of Mathematics

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The maximum angle condition of J. L. Synge was originally introduced in interpolation theory and further used in finite element analysis and applications for triangular and later also for tetrahedral finite element meshes. In this paper we present some of its generalizations to higher-dimensional simplicial elements. In particular, we prove optimal interpolation properties of linear simplicial elements in $ℝ^{d}$ that degenerate in some way.

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.

The Hardy-Lorentz spaces $H^{p, q} (ℝ ⁿ)$

Wael Abu-Shammala, Alberto Torchinsky (2007)

Studia Mathematica

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We deal with the Hardy-Lorentz spaces $H^{p, q} (ℝ ⁿ)$ where 0 < p ≤ 1, 0 < q ≤ ∞. We discuss the atomic decomposition of the elements in these spaces, their interpolation properties, and the behavior of singular integrals and other operators acting on them.

Interpolation theorem for the p-harmonic transform

Luigi D&#039;Onofrio, Tadeusz Iwaniec (2003)

Studia Mathematica

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We establish an interpolation theorem for a class of nonlinear operators in the Lebesgue spaces $ℒ^{s} (ℝ ⁿ)$ arising naturally in the study of elliptic PDEs. The prototype of those PDEs is the second order p-harmonic equation ${d i v | \nabla u |}^{p - 2 \nabla} u = d i v$ . In this example the p-harmonic transform is essentially inverse to $d i v (| \nabla |^{p - 2} \nabla)$ . To every vector field $\in ℒ^{q} (ℝ ⁿ, ℝ ⁿ)$ our operator $ℋ_{p}$ assigns the gradient of the solution, $ℋ_{p} = \nabla u \in ℒ^{p} (ℝ ⁿ, ℝ ⁿ)$ . The core of the matter is that we go beyond the natural domain of definition of this operator. Because of nonlinearity our...

Is $A^{- 1}$ an infinitesimal generator?

Hans Zwart (2007)

Banach Center Publications

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In this paper we study the question whether $A^{- 1}$ is the infinitesimal generator of a bounded C₀-semigroup if A generates a bounded C₀-semigroup. If the semigroup generated by A is analytic and sectorially bounded, then the same holds for the semigroup generated by $A^{- 1}$ . However, we construct a contraction semigroup with growth bound minus infinity for which $A^{- 1}$ does not generate a bounded semigroup. Using this example we construct an infinitesimal generator of a bounded semigroup for which its...

Conjugate norms in ℂⁿ and related geometrical problems

Baran Mirosław

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AbstractWe consider ℂⁿ as a normed space equipped with a complex norm F and we investigate some geometrical problems related with the notion of a conjugate norm F*. A crucial role in our considerations is played by the classical Shmul'yan theorem on exposed points in dual spaces. Many applications of this theorem are given for different problems including characterization of linear (biholomorphic) equivalence for a class of balls in ℂⁿ, calculation of the group of linear automorphisms...

Exact Kronecker constants of Hadamard sets

Kathryn E. Hare, L. Thomas Ramsey (2013)

Colloquium Mathematicae

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A set S of integers is called ε-Kronecker if every function on S of modulus one can be approximated uniformly to within ε by a character. The least such ε is called the ε-Kronecker constant, κ(S). The angular Kronecker constant is the unique real number α(S) ∈ [0,1/2] such that κ(S) = |exp(2πiα(S)) - 1|. We show that for integers m > 1 and d ≥ 1, $α 1, m, . . ., m^{d - 1} = (m^{d - 1} - 1) / (2 (m^{d} - 1))$ and α1,m,m²,... = 1/(2m).

On the Dirichlet problem associated with the Dunkl Laplacian

Mohamed Ben Chrouda (2016)

Annales Polonici Mathematici

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This paper deals with the questions of the existence and uniqueness of a solution to the Dirichlet problem associated with the Dunkl Laplacian $Δ_{k}$ as well as the hypoellipticity of $Δ_{k}$ on noninvariant open sets.