Results on Non-linear Approximation for Wavelet Bases in Weighted Function Spaces

2016 ◽  
pp. 145-189
Author(s):  
Eugenio Hernández ◽  
Maria de Natividade
Keyword(s):  
Linear Approximation ◽  
Function Spaces ◽  
Wavelet Bases ◽  
Weighted Function ◽  
Weighted Function Spaces ◽  
Non Linear

Wavelet bases and entropy numbers in weighted function spaces

2005 ◽  
Vol 278 (1-2) ◽  
pp. 108-132 ◽  
Author(s):  
Dorothee D. Haroske ◽  
Hans Triebel
Keyword(s):  
Function Spaces ◽  
Entropy Numbers ◽  
Wavelet Bases ◽  
Weighted Function ◽  
Weighted Function Spaces

scholarly journals Nuclear Embeddings in Weighted Function Spaces

2020 ◽  
Vol 92 (6) ◽  
Author(s):  
Dorothee D. Haroske ◽  
Leszek Skrzypczak
Keyword(s):  
Function Spaces ◽  
Operator Ideal ◽  
Weighted Spaces ◽  
Wavelet Bases ◽  
Weighted Function ◽  
Weighted Function Spaces ◽  
Diagonal Operators ◽  
Polynomial Type ◽  
Muckenhoupt Class

AbstractWe study nuclear embeddings for weighted spaces of Besov and Triebel–Lizorkin type where the weight belongs to some Muckenhoupt class and is essentially of polynomial type. Here we can extend our previous results concerning the compactness of corresponding embeddings. The concept of nuclearity was introduced by A. Grothendieck in 1955. Recently there is a refreshed interest to study such questions. This led us to the investigation in the weighted setting. We obtain complete characterisations for the nuclearity of the corresponding embedding. Essential tools are a discretisation in terms of wavelet bases, operator ideal techniques, as well as a very useful result of Tong about the nuclearity of diagonal operators acting in $$\ell _p$$ ℓ p spaces. In that way we can further contribute to the characterisation of nuclear embeddings of function spaces on domains.

scholarly journals Linear Differential Equations on Some Classes of Weighted Function Spaces

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1113
Author(s):  
Ahmed El-Sayed Ahmed ◽  
Amnah E. Shammaky
Keyword(s):  
Specific Growth ◽  
Function Spaces ◽  
Entire Solutions ◽  
Linear Type ◽  
Weighted Function ◽  
Weighted Function Spaces ◽  
Type Classes ◽  
Type Differential Equation ◽  
Linear Differential ◽  
Weighted Type

Some weighted-type classes of holomorphic function spaces were introduced in the current study. Moreover, as an application of the new defined classes, the specific growth of certain entire-solutions of a linear-type differential equation by the use of concerned coefficients of certain analytic-type functions, that is the equation h(k)+Kk−1(υ)h(k−1)+…+K1(υ)h′+K0(υ)h=0, will be discussed in this current research, whereas the considered coefficients K0(υ),…,Kk−1(υ) are holomorphic in the disc ΓR={υ∈C:|υ|<R},0<R≤∞. In addition, some non-trivial specific examples are illustrated to clear the roles of the obtained results with some sharpness sense. Hence, the obtained results are strengthen to some previous interesting results from the literature.

Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients

2019 ◽  
Vol 53 (5) ◽  
pp. 1507-1552 ◽  
Author(s):  
L. Herrmann ◽  
C. Schwab
Keyword(s):  
Monte Carlo ◽  
Random Field ◽  
Gaussian Random Field ◽  
Function Spaces ◽  
Monte Carlo Integration ◽  
Superior Performance ◽  
Elliptic Pdes ◽  
Weighted Function ◽  
Weighted Function Spaces ◽  
Quasi Monte Carlo

We analyze the convergence rate of a multilevel quasi-Monte Carlo (MLQMC) Finite Element Method (FEM) for a scalar diffusion equation with log-Gaussian, isotropic coefficients in a bounded, polytopal domain D ⊂ ℝd. The multilevel algorithm QL* which we analyze here was first proposed, in the case of parametric PDEs with sequences of independent, uniformly distributed parameters in Kuo et al. (Found. Comput. Math. 15 (2015) 411–449). The random coefficient is assumed to admit a representation with locally supported coefficient functions, as arise for example in spline- or multiresolution representations of the input random field. The present analysis builds on and generalizes our single-level analysis in Herrmann and Schwab (Numer. Math. 141 (2019) 63–102). It also extends the MLQMC error analysis in Kuo et al. (Math. Comput. 86 (2017) 2827–2860), to locally supported basis functions in the representation of the Gaussian random field (GRF) in D, and to product weights in QMC integration. In particular, in polytopal domains D ⊂ ℝd, d=2,3, our analysis is based on weighted function spaces to describe solution regularity with respect to the spatial coordinates. These spaces allow GRFs and PDE solutions whose realizations become singular at edges and vertices of D. This allows for non-stationary GRFs whose covariance operators and associated precision operator are fractional powers of elliptic differential operators in D with boundary conditions on ∂D. In the weighted function spaces in D, first order, Lagrangian Finite Elements on regular, locally refined, simplicial triangulations of D yield optimal asymptotic convergence rates. Comparison of the ε-complexity for a class of Matérn-like GRF inputs indicates, for input GRFs with low sample regularity, superior performance of the present MLQMC-FEM with locally supported representation functions over alternative representations, e.g. of Karhunen–Loève type. Our analysis yields general bounds for the ε-complexity of the MLQMC algorithm, uniformly with respect to the dimension of the parameter space.

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