scholarly journals Lifting of Quaternionic Frames to Higher Dimensions with Partial Ridges

2021 ◽  
Vol 31 (1) ◽  
Author(s):  
Florian Heinrich ◽  
Brigitte Forster
Keyword(s):  
Sobolev Spaces ◽  
Higher Dimensions ◽  
Embedding Theorems ◽  
Wavelet Frames ◽  
Skew Field ◽  
Novel Approach

AbstractWe consider the technique of lifting frames to higher dimensions with the ridge idea that originally was introduced by Grafakos and Sansing. We pursue a novel approach with regard to a non-commutative setting, concretely the skew-field of quaternions. Moreover, we allow for splitting dimensions and for lifting with regard to multi-ridges. To this end, we introduce quaternionic Sobolev spaces and prove the corresponding embedding theorems. We mention as concrete examples quaternionic wavelet frames and quaternionic shearlet frames, and give the respective lifted families.

Sobolev Spaces

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Sobolev Spaces ◽  
Embedding Theorems ◽  
Approximation Numbers ◽  
Selection Of

This chapter presents a selection of some of the most important results in the theory of Sobolev spacesn. Special emphasis is placed on embedding theorems and the question as to whether an embedding map is compact or not. Some results concerning the k-set contraction nature of certain embedding maps are given, for both bounded and unbounded space domains: also the approximation numbers of embedding maps are estimated and these estimates used to classify the embeddings.

scholarly journals Sobolev spaces on Lie groups: Embedding theorems and algebra properties

2019 ◽  
Vol 276 (10) ◽  
pp. 3014-3050 ◽  
Author(s):  
Tommaso Bruno ◽  
Marco M. Peloso ◽  
Anita Tabacco ◽  
Maria Vallarino
Keyword(s):  
Lie Groups ◽  
Sobolev Spaces ◽  
Embedding Theorems ◽  
And Algebra

Nonuniform nonhomogeneous dual wavelet frames in Sobolev spaces in $$L^2({\mathbb {K}})$$

2020 ◽  
Vol 31 (7-8) ◽  
pp. 1145-1156 ◽  
Author(s):  
O. Ahmad ◽  
N. A. Sheikh ◽  
M. A. Ali
Keyword(s):  
Sobolev Spaces ◽  
Wavelet Frames ◽  
Dual Wavelet Frames ◽  
Dual Wavelet

The Information Optimal Algorithm Based on Poly-Scaled Wavelet Wraps and Wavelet Frames with Finite Support

2011 ◽  
Vol 204-210 ◽  
pp. 1759-1762
Author(s):  
Tong Qi Zhang
Keyword(s):  
Multiresolution Analysis ◽  
Operator Theory ◽  
Matrix Theory ◽  
Integral Transform ◽  
Optimal Algorithm ◽  
Higher Dimensions ◽  
Wavelet Frames ◽  
Finite Support ◽  
Multi Scale ◽  
Vector Valued

In this paper, we propose the notion of vector-valued multiresolution analysis and the vector-valued mutivariate wavelet wraps with multi-scale factor of spaceL2(Rn, Cv), which are ge- neralizations of multivariate wavelet wraps. An approach for designing a sort of biorthogonal vec- tor-valued wavelet wraps in higher dimensions is presented and their biorthogonality trait is charac- -terized by virtue of integral transform, matrix theory, and operator theory. Two biorthogonality formulas regarding these wavelet wraps are established.

scholarly journals Dualwavelet frames in Sobolev spaces on local fields of positive characteristic

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 2091-2099
Author(s):  
Ishtaq Ahmad ◽  
Neyaz Sheikh
Keyword(s):  
Sobolev Spaces ◽  
Local Field ◽  
Positive Characteristic ◽  
Local Fields ◽  
Wavelet Frames ◽  
The Past ◽  
Dual Wavelet Frames ◽  
Dual Wavelet ◽  
Field Of Positive Characteristic

Wavelet frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of engineering and science. In this article, we obtain the characterization of nonhomogeneous wavelet frames and nonhomogeneous dual wavelet frames in a Sobolev spaces on a local field of positive characteristic by means of a pair of equations.

scholarly journals Chaos Driven Evolutionary Algorithm: a Novel Approach for Optimization

2022 ◽  
Vol 16 ◽  
pp. 21-25
Author(s):  
Roman Senkerik ◽  
Michal Pluhacek ◽  
Zuzana Kominkova Oplatkova
Keyword(s):  
Differential Evolution ◽  
Evolutionary Algorithm ◽  
Chaotic System ◽  
Random Number ◽  
Random Number Generator ◽  
Dissipative System ◽  
Higher Dimensions ◽  
Test Function ◽  
Novel Approach ◽  
Pseudo Random Number Generator

This research deals with the initial investigations on the concept of a chaos-driven evolutionary algorithm Differential evolution. This paper is aimed at the embedding of simple two-dimensional chaotic system, which is Lozi map, in the form of chaos pseudo random number generator for Differential Evolution. The chaotic system of interest is the discrete dissipative system. Repeated simulations were performed on standard benchmark Schwefel’s test function in higher dimensions. Finally, the obtained results are compared with canonical Differential Evolution.

Multipliers Between Sobolev Spaces

1991 ◽  
Vol 34 (4) ◽  
pp. 465-473
Author(s):  
R. C. Fabec
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Higher Dimensions ◽  
Sufficient Condition

AbstractA sufficient condition for the boundedness of a multiplier from a Sobolev space of index t > 1 / 4 to one of opposite index — t is obtained. The condition relates the indices of the Sobolev spaces to which the multiplier belongs to the pairs of Sobolev spaces between which the multiplier is bounded. The result is applied to homogeneous multipliers and a description of these multipliers in this setting is presesented. Extensions to higher dimensions are indicated.

On a class of weighted Sobolev spaces and the minimization of quadratic forms

1978 ◽  
Vol 81 (1-2) ◽  
pp. 119-130
Author(s):  
Frans Penning ◽  
Niko Sauer
Keyword(s):  
Weight Function ◽  
Sobolev Spaces ◽  
Compact Support ◽  
Quadratic Forms ◽  
Weighted Sobolev Spaces ◽  
Weight Functions ◽  
Compact Embedding ◽  
Embedding Theorems ◽  
Smooth Functions ◽  
Square Integrability

SynopsisIn this paper a class of weighted Sobolev spaces defined in terms of square integrability of the gradient multiplied by a weight function, is studied. The domain of integration is either the spaceRnor a half-space ofRn. Conditions on the weight functions that will ensure density of classes of smooth functions or functions with compact support, and compact embedding theorems, are derived. Finally the results are applied to a class of isoperimetrical problems in the calculus of variations in which the domain of integration is unbounded.

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