Several notes on the circumradius condition

Václav Kučera

Displaying similar documents to “Several notes on the circumradius condition”

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

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

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.

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 $(ω_{0}, ω_{1})$ .

Real method of interpolation on subcouples of codimension one

S. V. Astashkin, P. Sunehag (2008)

Studia Mathematica

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We find necessary and sufficient conditions under which the norms of the interpolation spaces ${(N ₀, N ₁)}_{θ, q}$ and ${(X ₀, X ₁)}_{θ, q}$ are equivalent on N, where N is the kernel of a nonzero functional ψ ∈ (X₀ ∩ X₁)* and $N_{i}$ is the normed space N with the norm inherited from $X_{i}$ (i = 0,1). Our proof is based on reducing the problem to its partial case studied by Ivanov and Kalton, where ψ is bounded on one of the endpoint spaces. As an application we completely resolve the problem of when the range of the operator $T_{θ} = S - 2^{θ} I$ (S...

Modulus of dentability in $L ¹ + L^{\infty}$

Adam Bohonos, Ryszard Płuciennik (2008)

Banach Center Publications

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We introduce the notion of the modulus of dentability defined for any point of the unit sphere S(X) of a Banach space X. We calculate effectively this modulus for denting points of the unit ball of the classical interpolation space $L ¹ + L^{\infty} .$ Moreover, a criterion for denting points of the unit ball in this space is given. We also show that none of denting points of the unit ball of $L ¹ + L^{\infty}$ is a LUR-point. Consequently, the set of LUR-points of the unit ball of $L ¹ + L^{\infty}$ is empty.

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...

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...

$H^{\infty}$ 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.

Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces

Yi Liu, Wen Yuan (2017)

Czechoslovak Mathematical Journal

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Let $θ \in (0, 1)$ , $λ \in [0, 1)$ and $p, p_{0}, p_{1} \in (1, \infty]$ be such that $(1 - θ) / p_{0} + θ / p_{1} = 1 / p$ , and let $ϕ, ϕ_{0}, ϕ_{1}$ be some admissible functions such that $ϕ, ϕ_{0}^{p / p_{0}}$ and $ϕ_{1}^{p / p_{1}}$ are equivalent. We first prove that, via the $\pm$ interpolation method, the interpolation $〈 L_{ϕ_{0}}^{p_{0}), λ} (𝒳), L_{ϕ_{1}}^{p_{1}), λ} (𝒳), θ 〉$ of two generalized grand Morrey spaces on a quasi-metric measure space $𝒳$ is the generalized grand Morrey space $L_{ϕ}^{p), λ} (𝒳)$ . Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces.

Some remarks on the interpolation spaces $A^{θ}, A_{θ}$

Mohammad Daher (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $(A_{0}, A_{1})$ be a regular interpolation couple. Under several different assumptions on a fixed $A^{β}$ , we show that $A^{θ} = A_{θ}$ for every $θ \in (0, 1)$ . We also deal with assumptions on ${\bar{A}}^{β}$ , the closure of $A^{β}$ in the dual of ${(A_{0}^{*}, A_{1}^{*})}_{β}$ .

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

M. Ajzerle (1992)

Matematički Vesnik

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Stein’s method in high dimensions with applications

Adrian Röllin (2013)

Annales de l'I.H.P. Probabilités et statistiques

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Let $h$ be a three times partially differentiable function on $ℝ^{n}$ , let $X = (X_{1}, ..., X_{n})$ be a collection of real-valued random variables and let $Z = (Z_{1}, ..., Z_{n})$ be a multivariate Gaussian vector. In this article, we develop Stein’s method to give error bounds on the difference $𝔼 h (X) - 𝔼 h (Z)$ in cases where the coordinates of $X$ are not necessarily independent, focusing on the high dimensional case $n \to \infty$ . In order to express the dependency structure we use Stein couplings, which allows for a broad range of applications, such as classic...