Approximation from sparse grids and function spaces of dominating mixed smoothness

Winfried Sickel

Displaying similar documents to “Approximation from sparse grids and function spaces of dominating mixed smoothness”

Some $R^{N}$ capacities and applications - the lemma on mixed derivatives revisited

J. Korevaar (1986)

Matematički Vesnik

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Function spaces with dominating mixed smoothness

Jan Vybiral

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We study several techniques which are well known in the case of Besov and Triebel-Lizorkin spaces and extend them to spaces with dominating mixed smoothness. We use the ideas of Triebel to prove three important decomposition theorems. We deal with so-called atomic, subatomic and wavelet decompositions. All these theorems have much in common. Roughly speaking, they say that a function f belongs to some function space (say $S_{p, q}^{r ̅} A$ ) if, and only if, it can be decomposed as $f (x) = \sum_{ν} \sum_{m} λ_{ν m} a_{ν m} (x)$ , convergence in S’, with...

On the existence of exactly one solution of integral equations in the space $L^{P}$ with a mixed norm

Bogdan Rzepecki (1975)

Annales Polonici Mathematici

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Variations of mixed Hodge structure attached to the deformation theory of a complex variation of Hodge structures

Philippe Eyssidieux, Carlos Simpson (2011)

Journal of the European Mathematical Society

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Let $X$ be a compact Kähler manifold, $x \in X$ be a base point and $ρ : π_{1} (X, x) \to G L_{N} (C)$ be the monodromy representation of a $𝒞$ -VHS. Building on Goldman–Millson’s classical work, we construct a mixed Hodge structure on the complete local ring of the representation variety at $ρ$ and a variation of mixed Hodge structures whose monodromy is the universal deformation of $ρ$ .

An example of the equation $u_{t} = u_{x x} + f (x, t, u)$ with distinct maximum and minimum solutions of a mixed problem

W. Mlak (1963)

Annales Polonici Mathematici

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ℓ¹-Spreading models in subspaces of mixed Tsirelson spaces

Denny H. Leung, Wee-Kee Tang (2006)

Studia Mathematica

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We investigate the existence of higher order ℓ¹-spreading models in subspaces of mixed Tsirelson spaces. For instance, we show that the following conditions are equivalent for the mixed Tsirelson space $X = T [{(θ ₙ, ₙ)}_{n = 1}^{\infty}]$ : (1) Every block subspace of X contains an $ℓ ¹ -_{ω}$ -spreading model, (2) The Bourgain ℓ¹-index $I_{b} (Y) = I (Y) > ω^{ω}$ for any block subspace Y of X, (3) $l i m ₘ l i m s u p ₙ θ_{m + n} / θ ₙ > 0$ and every block subspace Y of X contains a block sequence equivalent to a subsequence of the unit vector basis of X. Moreover, if one (and hence all) of these...

$L_{p}$ - and $S_{p, q}^{r} B$ -discrepancy of (order 2) digital nets

Lev Markhasin (2015)

Acta Arithmetica

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Dick proved that all dyadic order 2 digital nets satisfy optimal upper bounds on the $L_{p}$ -discrepancy. We prove this for arbitrary prime base b with an alternative technique using Haar bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds on the discrepancy function in Besov spaces with dominating mixed smoothness for a certain parameter range, and enlarge that range for order 2 digital nets. The discrepancy function in Triebel-Lizorkin and Sobolev spaces with...

The diagonal mapping in mixed norm spaces

Guangbin Ren, Jihuai Shi (2004)

Studia Mathematica

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For any holomorphic function F in the unit polydisc Uⁿ of ℂⁿ, we consider its restriction to the diagonal, i.e., the function in the unit disc U of ℂ defined by F(z) = F(z,...,z), and prove that the diagonal mapping maps the mixed norm space $H^{p, q, α} (U ⁿ)$ of the polydisc onto the mixed norm space $H^{p, q, | α | + (p / q + 1) (n - 1)} (U)$ of the unit disc for any 0 < p < ∞ and 0 < q ≤ ∞.

Noncharacteristic mixed problems for hyperbolic systems of the first order

Ewa Zadrzyńska

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CONTENTS1. Introduction....................................................................................................................................................52. Notations and preliminaries .........................................................................................................................11 2.1. Function spaces and spaces of distributions............................................................................................11 2.2. Perturbations...

$L_{\infty}$ -Convergence of Finite Element Approximation

J. A. Nitsche (1975)

Publications mathématiques et informatique de Rennes

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A correction to “Some mean convergence and complete convergence theorems for sequences of $m$ -linearly negative quadrant dependent random variables”

Yongfeng Wu, Andrew Rosalsky, Andrei Volodin (2017)

Applications of Mathematics

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The authors provide a correction to “Some mean convergence and complete convergence theorems for sequences of $m$ -linearly negative quadrant dependent random variables”.

Pointwise convergence of nonconventional averages

I. Assani (2005)

Colloquium Mathematicae

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We answer a question of H. Furstenberg on the pointwise convergence of the averages $1 / N \sum_{n = 1}^{N} U ⁿ (f \cdot R ⁿ (g))$ , where U and R are positive operators. We also study the pointwise convergence of the averages $1 / N \sum_{n = 1}^{N} f (S ⁿ x) g (R ⁿ x)$ when T and S are measure preserving transformations.

Mixed $A_{p} - A_{\infty}$ estimates with one supremum

Andrei K. Lerner, Kabe Moen (2013)

Studia Mathematica

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We establish several mixed $A_{p} - A_{\infty}$ bounds for Calderón-Zygmund operators that only involve one supremum. We address both cases when the $A_{\infty}$ part of the constant is measured using the exponential-logarithmic definition and using the Fujii-Wilson definition. In particular, we answer a question of the first author and provide an answer, up to a logarithmic factor, to a conjecture of Hytönen and Lacey. Moreover, we give an example to show that our bounds with the logarithmic factors can be arbitrarily...

On Clifford-type structures

Wiesław Królikowski

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On the rate of convergence of difference schemes for the heat transfer equation on the solutions from $w_{2}^{s, s / 2}$

L. D. Ivanović, B. S. Jovanović, E. E. Süli (1984)

Matematički Vesnik

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On the oscillation of forced second order mixed-nonlinear elliptic equations

Zhiting Xu (2010)

Annales Polonici Mathematici

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Oscillation theorems are established for forced second order mixed-nonlinear elliptic differential equations ⎧ ${d i v (A (x) | | \nabla y | |}^{p - 1} {\nabla y) + ⟨ b (x), | | \nabla y | |}^{p - 1} \nabla y ⟩ + C (x, y) = e (x)$ , ⎨ ⎩ $C (x, y) = {c (x) | y |}^{p - 1} y + \sum_{i = 1}^{m} c_{i} (x) {| y |}^{p_{i} - 1} y$ under quite general conditions. These results are extensions of the recent results of Sun and Wong, [J. Math. Anal. Appl. 334 (2007)] and Zheng, Wang and Han [Appl. Math. Lett. 22 (2009)] for forced second order ordinary differential equations with mixed nonlinearities, and include some known oscillation results in the literature

Minimality properties of Tsirelson type spaces

Denka Kutzarova, Denny H. Leung, Antonis Manoussakis, Wee-Kee Tang (2008)

Studia Mathematica

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We study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis $(e_{k})$ is said to be subsequentially minimal if for every normalized block basis $(x_{k})$ of $(e_{k})$ , there is a further block basis $(y_{k})$ of $(x_{k})$ such that $(y_{k})$ is equivalent to a subsequence of $(e_{k})$ . Sufficient conditions are given for a partly modified mixed Tsirelson space to be subsequentially minimal, and connections with Bourgain’s ℓ¹-index are established. It is also shown that a large class of...

Convergence in the dual of certain $K M_{p}$ -spaces

Larry Kitchens, Charles Swartz (1974)

Colloquium Mathematicae

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Lagrange approximation in Banach spaces

Lisa Nilsson, Damián Pinasco, Ignacio M. Zalduendo (2015)

Czechoslovak Mathematical Journal

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Starting from Lagrange interpolation of the exponential function $e^{z}$ in the complex plane, and using an integral representation formula for holomorphic functions on Banach spaces, we obtain Lagrange interpolating polynomials for representable functions defined on a Banach space $E$ . Given such a representable entire funtion $f : E \to ℂ$ , in order to study the approximation problem and the uniform convergence of these polynomials to $f$ on bounded sets of $E$ , we present a sufficient growth condition on...

A note on necessary and sufficient conditions for convergence of the finite element method

Kučera, Václav

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In this short note, we present several ideas and observations concerning finite element convergence and the role of the maximum angle condition. Based on previous work, we formulate a hypothesis concerning a necessary condition for $O (h)$ convergence and show a simple relation to classical problems in measure theory and differential geometry which could lead to new insights in the area.

Linearized plasticity is the evolutionary $Γ$ -limit of finite plasticity

Alexander Mielke, Ulisse Stefanelli (2013)

Journal of the European Mathematical Society

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We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via $Γ$ -convergence for rate-independent processes that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity.

On the completeness of $ℒ_{p}$ -spaces over a charge

Sreela Gangopadhyay (1990)

Colloquium Mathematicae

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On the rate of convergence of the Bézier-type operators

Grażyna Anioł (2006)

Bollettino dell'Unione Matematica Italiana

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For bounded functions $f$ on an interval $I$ , in particular, for functions of bounded p-th power variation on $I$ there is estimated the rate of pointwise convergence of the Bezier-type modification of the discrete Feller operators. In the main theorem the Chanturiya modulus of variation is used.

Subspaces $H_{E}^{p, q, α}$ of mixed-norm spaces $H^{p, q, α}$ of holomorphic functions

M. Jevtić (1985)

Matematički Vesnik

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