The Approximation Lemma

1978 ◽  
pp. 37-51
Author(s):  
R. V. Gamkrelidze
Keyword(s):  
Approximation Lemma

scholarly journals Refinements of the holonomic approximation lemma

2018 ◽  
Vol 18 (4) ◽  
pp. 2265-2303 ◽  
Author(s):  
Daniel Álvarez-Gavela
Keyword(s):  
Approximation Lemma

scholarly journals On locally essentially bounded divergence measure fields and sets of locally finite perimeter

2020 ◽  
Vol 13 (2) ◽  
pp. 179-217 ◽  
Author(s):  
Giovanni E. Comi ◽  
Kevin R. Payne
Keyword(s):  
Approximation Theory ◽  
Smooth Boundary ◽  
Direct Proof ◽  
Leibniz Rule ◽  
Divergence Measure ◽  
Locally Finite ◽  
Sets Of Finite Perimeter ◽  
Finite Perimeter ◽  
Approximation Lemma ◽  
Natural Way

AbstractChen, Torres and Ziemer ([9], 2009) proved the validity of generalized Gauss–Green formulas and obtained the existence of interior and exterior normal traces for essentially bounded divergence measure fields on sets of finite perimeter using an approximation theory through sets with a smooth boundary. However, it is known that the proof of a crucial approximation lemma contained a gap. Taking inspiration from a previous work of Chen and Torres ([7], 2005) and exploiting ideas of Vol’pert ([29], 1985) for essentially bounded fields with components of bounded variation, we present here a direct proof of generalized Gauss–Green formulas for essentially bounded divergence measure fields on sets of finite perimeter which includes the existence and essential boundedness of the normal traces. Our approach appears to be simpler since it does not require any special approximation theory for the domains and it relies only on the Leibniz rule for divergence measure fields. This freedom allows one to localize the constructions and to derive more general statements in a natural way.

The fuzzy neural network approximation lemma

1999 ◽  
Vol 102 (2) ◽  
pp. 227-236 ◽  
Author(s):  
Thomas Feuring ◽  
Wolfram-M. Lippe
Keyword(s):  
Neural Network ◽  
Fuzzy Neural Network ◽  
Fuzzy Neural ◽  
Approximation Lemma

scholarly journals An hp-Hybrid High-Order Method for Variable Diffusion on General Meshes

2017 ◽  
Vol 17 (3) ◽  
pp. 359-376 ◽  
Author(s):  
Joubine Aghili ◽  
Daniele A. Di Pietro ◽  
Berardo Ruffini
Keyword(s):  
Space Dimension ◽  
Diffusion Problem ◽  
High Order ◽  
Local Anisotropy ◽  
Arbitrary Space ◽  
Approximation Lemma ◽  
Variable Diffusion ◽  
Convergence Estimates ◽  
General Meshes ◽  
Polytopal Meshes

AbstractIn this work, we introduce and analyze anhp-hybrid high-order (hp-HHO) method for a variable diffusion problem. The proposed method is valid in arbitrary space dimension and for fairly general polytopal meshes. Variable approximation degrees are also supported. We provehp-convergence estimates for both the energy- andL^{2}-norms of the error, which are the first of this kind for Hybrid High-Order methods. These results hinge on a novelhp-approximation lemma valid for general polytopal elements in arbitrary space dimension. The estimates are additionally fully robust with respect to the heterogeneity of the diffusion coefficient, and show only a mild dependence on the square root of the local anisotropy, improving previous results for HHO methods. The expected exponential convergence behavior is numerically demonstrated on a variety of meshes for both isotropic and strongly anisotropic diffusion problems.

Some Extensions of a Theorem of Hardy, Littlewood and Pólya and Their Applications

1974 ◽  
Vol 26 (6) ◽  
pp. 1321-1340 ◽  
Author(s):  
Kong-Ming Chong
Keyword(s):  
Convex Functions ◽  
Finite Interval ◽  
Order Relation ◽  
Spectral Order ◽  
Measurable Functions ◽  
Approximation Lemma

In [6], by means of convex functions Φ :R→R, Hardy, Littlewood and Pólya proved a theorem characterizing the strong spectral order relation for any two measurable functions which are defined on a finite interval and which they implicitly assumed to be essentially bounded (cf. [6, the approximation lemma on p. 150 and Theorem 9 on p. 151 of their paper]; see also L. Mirsky [10, pp. 328-329] and H. D. Brunk [1,Theorem A, p. 820]).

BEST CONSTANTS AND POHOZAEV IDENTITY FOR HARDY–SOBOLEV-TYPE OPERATORS

2013 ◽  
Vol 15 (03) ◽  
pp. 1250050 ◽  
Author(s):  
ADIMURTHI
Keyword(s):  
Hardy Inequality ◽  
Direct Proof ◽  
Sobolev Type ◽  
General Elliptic ◽  
Functional Aspects ◽  
Approximation Lemma ◽  
Definition Of ◽  
Multiparticle Systems ◽  
The Hardy Inequality ◽  
Optimal Inequalities

This paper is threefold. Firstly, we reformulate the definition of the norm induced by the Hardy inequality (see [J. L. Vázquez and N. B. Zographopoulos, Functional aspects of the Hardy inequality. Appearance of a hidden energy, preprint (2011); http://arxiv. org/abs/1102.5661]) to more general elliptic and sub-elliptic Hardy–Sobolev-type operators. Secondly, we derive optimal inequalities (see [C. Cowan, Optimal inequalities for general elliptic operator with improvement, Commun. Pure Appl. Anal.9(1) (2010) 109–140; N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy–Rellich inequalities, Math. Ann.349(1) (2010) 1–57 (electronic)]) for multiparticle systems in ℝN and Heisenberg group. In particular, we provide a direct proof of an optimal inequality with multipolar singularities shown in [R. Bossi, J. Dolbeault and M. J. Esteban, Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators, Commun. Pure Appl. Anal.7(3) (2008) 533–562]. Finally, we prove an approximation lemma which allows to show that the domain of the Dirichlet–Laplace operator is dense in the domain of the corresponding Hardy operators. As a consequence, in some particular cases, we justify the Pohozaev-type identity for such operators.

scholarly journals A sparse regular approximation lemma

2019 ◽  
Vol 371 (10) ◽  
pp. 6779-6814 ◽  
Author(s):  
Guy Moshkovitz ◽  
Asaf Shapira
Keyword(s):  
Regular Approximation ◽  
Approximation Lemma
Sign in / Sign up

Export Citation Format

Share Document