Journal•ISSN: 0036-1410
Siam Journal on Mathematical Analysis
Society for Industrial and Applied Mathematics
About: Siam Journal on Mathematical Analysis is an academic journal published by Society for Industrial and Applied Mathematics. The journal publishes majorly in the area(s): Boundary value problem & Nonlinear system. It has an ISSN identifier of 0036-1410. Over the lifetime, 5393 publications have been published receiving 178079 citations. The journal is also known as: Journal on mathematical analysis & Society for Industrial and Applied Mathematics journal on mathematical analysis.
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TL;DR: In this article, the authors studied square integrable coefficients of an irreducible representation of the non-unimodular $ax + b$-group and obtained explicit expressions in the case of a particular analyzing family that plays a role analogous to coherent states (Gabor wavelets) in the usual $L_2 $ -theory.
Abstract: An arbitrary square integrable real-valued function (or, equivalently, the associated Hardy function) can be conveniently analyzed into a suitable family of square integrable wavelets of constant shape, (i.e. obtained by shifts and dilations from any one of them.) The resulting integral transform is isometric and self-reciprocal if the wavelets satisfy an “admissibility condition” given here. Explicit expressions are obtained in the case of a particular analyzing family that plays a role analogous to that of coherent states (Gabor wavelets) in the usual $L_2 $ -theory. They are written in terms of a modified $\Gamma $-function that is introduced and studied. From the point of view of group theory, this paper is concerned with square integrable coefficients of an irreducible representation of the nonunimodular $ax + b$-group.
3,423 citations
TL;DR: In this article, the authors define a notion of "two-scale" convergence, which is aimed at a better description of sequences of oscillating functions, and prove that bounded sequences in $L^2 (Omega )$ are relatively compact with respect to this new type of convergence.
Abstract: Following an idea of G. Nguetseng, the author defines a notion of “two-scale” convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in $L^2 (\Omega )$ are proven to be relatively compact with respect to this new type of convergence. A corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its “two-scale” limit, up to a strongly convergent remainder in $L^2 (\Omega )$) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the so-called energy method of Tartar. The power and simplicity of the two-scale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear second-order elliptic equations.
2,279 citations
TL;DR: The lifting wavelet as discussed by the authors is a simple construction of second generation wavelets that can be adapted to intervals, domains, surfaces, weights, and irregular samples, and it leads to a faster, in-place calculation of the wavelet transform.
Abstract: We present the lifting scheme, a simple construction of second generation wavelets; these are wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, in-place calculation of the wavelet transform. Several examples are included.
2,082 citations
TL;DR: The Fokker-Planck equation as mentioned in this paper describes the evolution of the probability density for a stochastic process associated with an Ito Stochastic Differential Equation.
Abstract: The Fokker--Planck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It ...
1,552 citations
TL;DR: In this paper, the convergence of the functional π (x,y) with support in a fixed compact set is studied and a convergence theorem is proved under the weaker assumption that π(x, y) remains in a bounded subset of the set, and the use of multiple scale expansions in homogenization is justified.
Abstract: The convergence, as $\varepsilon \downarrow 0$, of the functional $F_\varepsilon (\Psi ) = \int _{\mathbb{R}^N } u_\varepsilon (x)\Psi (x,{x / \varepsilon })$ associated with a given $L^2 $ function $u_\varepsilon $ with support in a fixed compact set is studied. The test functions $\Psi (x,y)$ are continuous on $\mathbb{R}^N \times \mathbb{R}^N $ and periodic in y. A convergence theorem is proved under the weaker assumption that $u_\varepsilon $ remains in a bounded subset of $L^2 $. Finally, the use of multiple-scale expansions in homogenization is justified, and a new approach is proposed for the mathematical analysis of homogenization problems.
1,422 citations