scholarly journals Homogeneous Besov Spaces on Stratified Lie Groups and Their Wavelet Characterization

2012 ◽  
Vol 2012 ◽  
pp. 1-41 ◽  
Author(s):  
Hartmut Führ ◽  
Azita Mayeli
Keyword(s):  
Lie Groups ◽  
Besov Spaces ◽  
Wavelet Transforms ◽  
Discrete Wavelet ◽  
Wavelet Frames ◽  
Euclidean Case ◽  
Continuous Wavelet Transforms ◽  
Continuous Scale ◽  
Stratified Lie Groups ◽  
Homogeneous Besov Space

We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both in terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov spaceB˙p,qsin terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield the same Besov spaces. We then turn to wavelet characterizations, first via continuous wavelet transforms (which can be viewed as continuous-scale Littlewood-Paley decompositions), then via discretely indexed systems. We prove the existence of wavelet frames and associated atomic decomposition formulas for all homogeneous Besov spacesB˙p,qswith1≤p,q<∞ands∈ℝ.

scholarly journals A discrete wavelet analysis of freak waves in the ocean

2004 ◽  
Vol 2004 (5) ◽  
pp. 379-394 ◽  
Author(s):  
En-Bing Lin ◽  
Paul C. Liu
Keyword(s):  
Wavelet Analysis ◽  
Wavelet Transforms ◽  
Discrete Wavelet ◽  
Continuous Wavelet ◽  
Freak Waves ◽  
Freak Wave ◽  
Continuous Wavelet Transforms ◽  
Basic Properties ◽  
Discrete Wavelet Analysis

A freak wave is a wave of very considerable height, ahead of which there is a deep trough. A case study examines some basic properties developed by performing wavelet analysis on a freak wave. We demonstrate several applications of wavelets and discrete and continuous wavelet transforms on the study of a freak wave. A modeling setting for freak waves will also be mentioned.

scholarly journals Wavelet Frames for Distributions in Anisotropic Besov Spaces

2005 ◽  
Vol 12 (4) ◽  
pp. 637-658
Author(s):  
Dorothee D. Haroske ◽  
Erika Tamási
Keyword(s):  
Large Class ◽  
Besov Spaces ◽  
Studia Math ◽  
Wavelet Frames ◽  
Anisotropic Besov Spaces ◽  
Wavelet Decompositions

Abstract This paper deals with wavelet frames in anisotropic Besov spaces , 𝑠 ∈ ℝ, 0 < 𝑝, 𝑞 ≤ ∞, and 𝑎 = (𝑎1, . . . , 𝑎𝑛) is an anisotropy, with 𝑎𝑖 > 0, 𝑖 = 1, . . . , 𝑛, 𝑎1 + . . . + 𝑎𝑛 = 𝑛. We present sub-atomic and wavelet decompositions for a large class of distributions. To some extent our results can be regarded as anisotropic counterparts of those recently obtained in [Triebel, Studia Math. 154: 59–88, 2003].

Weighted higher order exponential type inequalities in metric spaces and applications

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huiju Wang ◽  
Pengcheng Niu
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Sobolev Spaces ◽  
Exponential Type ◽  
Metric Spaces ◽  
Type Inequality ◽  
Higher Order ◽  
Geodesic Space ◽  
Stratified Lie Groups ◽  
Weighted Case

AbstractIn this paper, we establish weighted higher order exponential type inequalities in the geodesic space {({X,d,\mu})} by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given.

A New Method for Characterizing Replacement Rate Variation in Molecular Sequences: Application of the Fourier and Wavelet Models to Drosophila and Mammalian Proteins

Genetics ◽  
2000 ◽  
Vol 154 (1) ◽  
pp. 381-395
Author(s):  
Pavel Morozov ◽  
Tatyana Sitnikova ◽  
Gary Churchill ◽  
Francisco José Ayala ◽  
Andrey Rzhetsky
Keyword(s):  
Wavelet Transforms ◽  
Parametric Bootstrap ◽  
Real Data ◽  
New Method ◽  
Rate Variation ◽  
Discrete Wavelet ◽  
Data Sets ◽  
Ratio Test ◽  
Replacement Rate ◽  
New Models

Abstract We propose models for describing replacement rate variation in genes and proteins, in which the profile of relative replacement rates along the length of a given sequence is defined as a function of the site number. We consider here two types of functions, one derived from the cosine Fourier series, and the other from discrete wavelet transforms. The number of parameters used for characterizing the substitution rates along the sequences can be flexibly changed and in their most parameter-rich versions, both Fourier and wavelet models become equivalent to the unrestricted-rates model, in which each site of a sequence alignment evolves at a unique rate. When applied to a few real data sets, the new models appeared to fit data better than the discrete gamma model when compared with the Akaike information criterion and the likelihood-ratio test, although the parametric bootstrap version of the Cox test performed for one of the data sets indicated that the difference in likelihoods between the two models is not significant. The new models are applicable to testing biological hypotheses such as the statistical identity of rate variation profiles among homologous protein families. These models are also useful for determining regions in genes and proteins that evolve significantly faster or slower than the sequence average. We illustrate the application of the new method by analyzing human immunoglobulin and Drosophilid alcohol dehydrogenase sequences.

scholarly journals The Higher Order Riesz Transform andBMOType Space Associated with Schrödinger Operators on Stratified Lie Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Yu Liu ◽  
Jianfeng Dong
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Riesz Transform ◽  
Integral Operators ◽  
Higher Order ◽  
Reverse Hölder Class ◽  
Stratified Lie Group ◽  
Homogeneous Dimension ◽  
Stratified Lie Groups ◽  
Reverse Hölder

Assume thatGis a stratified Lie group andQis the homogeneous dimension ofG. Let-Δbe the sub-Laplacian onGandW≢0a nonnegative potential belonging to certain reverse Hölder classBsfors≥Q/2. LetL=-Δ+Wbe a Schrödinger operator on the stratified Lie groupG. In this paper, we prove the boundedness of some integral operators related toL, such asL-1∇2,L-1W, andL-1(-Δ) on the spaceBMOL(G).

COMPARATIVE ANALYSIS OF SPECTRA OF THE 2000-YEAR NORTHERN HEMISPHERE AVERAGE SURFACE AIR TEMPERATURE RECONSTRUCTIONS

2021 ◽  
pp. 5-13
Author(s):  
N. M. DATSENKO ◽  
◽  
D. M. SONECHKIN ◽  
B. YANG ◽  
J.-J. LIU ◽  
...  
Keyword(s):  
High Resolution ◽  
Northern Hemisphere ◽  
Air Temperature ◽  
Wavelet Transforms ◽  
Spectral Composition ◽  
Low Frequency ◽  
Surface Air Temperature ◽  
Temporal Variations ◽  
Continuous Wavelet Transforms ◽  
Stable Estimation

The spectral composition of temporal variations in the Northern Hemisphere mean surface air temperature is estimated and compared in 2000-year paleoclimatic reconstructions. Continuous wavelet transforms of these reconstructions are used for the stable estimation of energy spectra. It is found that low-frequency parts of the spectra (the periods of temperature variations of more than 100 years) based on such high-resolution paleoclimatic indicators as tree rings, corals, etc., are similar to the spectrum of white noise, that is never observed in nature. This seems unrealistic. The famous reconstruction called “Hockey Stick” is among such unrealistic reconstructions. Reconstructions based not only on high-resolution but also on low-resolution indicators seem to be more realistic, since the low-frequency parts of their spectra have the pattern of red noise. They include the “Boomerang” reconstruction showing that some warm periods close to the present-day one were observed in the past.

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