Hausdorff dimension of a fractal interpolation function

Guantie Deng

Displaying similar documents to “Hausdorff dimension of a fractal interpolation function”

A counterexample to the Γ-interpolation conjecture

Adama S. Kamara (2015)

Annales Polonici Mathematici

Similarity:

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.

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

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

Annales Polonici Mathematici

Similarity:

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.

Lions-Peetre reiteration formulas for triples and their applications

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

Studia Mathematica

Similarity:

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

Similarity:

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

Similarity:

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

Weak-type operators and the strong fundamental lemma of real interpolation theory

N. Krugljak, Y. Sagher, P. Shvartsman (2005)

Studia Mathematica

Similarity:

We prove an interpolation theorem for weak-type operators. This is closely related to interpolation between weak-type classes. Weak-type classes at the ends of interpolation scales play a similar role to that played by BMO with respect to the $L^{p}$ interpolation scale. We also clarify the roles of some of the parameters appearing in the definition of the weak-type classes. The interpolation theorem follows from a K-functional inequality for the operators, involving the Calderón operator....

Fractal Interpolation in Creating Prefractal Images

Ljubiša M. Kocić, Alba Chiara Simoncelli (2000)

Visual Mathematics

Similarity:

On interpolation error on degenerating prismatic elements

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

Applications of Mathematics

Similarity:

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.

A functional calculus description of real interpolation spaces for sectorial operators

Markus Haase (2005)

Studia Mathematica

Similarity:

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.

Interpolation of quasicontinuous functions

Joan Cerdà, Joaquim Martín, Pilar Silvestre (2011)

Banach Center Publications

Similarity:

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

Spherical basis function approximation with particular trend functions

Segeth, Karel

Similarity:

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 complex interpolation and spectral continuity

Karen Saxe (1998)

Studia Mathematica

Similarity:

Let ${[X_{0}, X_{1}]}_{t}$ , 0 ≤ t ≤ 1, be Banach spaces obtained via complex interpolation. With suitable hypotheses, linear operators T that act boundedly on both $X_{0}$ and $X_{1}$ will act boundedly on each ${[X_{0}, X_{1}]}_{t}$ . Let $T_{t}$ denote such an operator when considered on ${[X_{0}, X_{1}]}_{t}$ , and $σ (T_{t})$ denote its spectrum. We are motivated by the question of whether or not the map $t \to σ (T_{t})$ is continuous on (0,1); this question remains open. In this paper, we study continuity of two related maps: $t \to {(σ (T_{t}))}^{\land}$ (polynomially convex hull) and $t \to \partial_{e} (σ (T_{t}))$ (boundary of the polynomially...

Interpolation methods of means and orbits

Mieczysław Mastyło (2005)

Studia Mathematica

Similarity:

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

Charles E. C Cleaver (1973)

Colloquium Mathematicae

Similarity:

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

Antti Hannukainen, Sergey Korotov, Michal Křížek (2017)

Applications of Mathematics

Similarity:

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

Similarity:

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.

Об $(H^{p}, l^{r})$ -интерполяционных последовательностях в единичном круге

С.В. Шведенко (1982)

Matematiceskij sbornik

Similarity:

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

M. Ajzerle (1992)

Matematički Vesnik

Similarity:

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

Giovanni Dore (1999)

Studia Mathematica

Similarity:

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.

Several notes on the circumradius condition

Václav Kučera (2016)

Applications of Mathematics

Similarity:

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...

Extending n-convex functions

Allan Pinkus, Dan Wulbert (2005)

Studia Mathematica

Similarity:

We are given data α₁,..., αₘ and a set of points E = x₁,...,xₘ. We address the question of conditions ensuring the existence of a function f satisfying the interpolation conditions $f (x_{i}) = α_{i}$ , i = 1,...,m, that is also n-convex on a set properly containing E. We consider both one-point extensions of E, and extensions to all of ℝ. We also determine bounds on the n-convex functions satisfying the above interpolation conditions.