Showing papers in "Journal of Fourier Analysis and Applications in 1998"
TL;DR: In this paper, a self-contained derivation from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering, is presented, which asymptotically reduces the computational complexity of the transform by a factor two.
Abstract: This article is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formulaSL(n;R[z, z−1])=E(n;R[z, z−1])); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e., non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers.
2,357 citations
437 citations
82 citations
TL;DR: In this article, the authors give general mathematical results concerning oscillating singularities and study examples of functions composed only of OSSs, which are defined by explicit coefficients on an orthonormal wavelet basis.
Abstract: We give general mathematical results concerning oscillating singularities and we study examples of functions composed only of oscillating singularities. These functions are defined by explicit coefficients on an orthonormal wavelet basis. We compute their Holder regularity and oscillation at every point and we deduce their spectrum of oscillating singularities.
78 citations
TL;DR: In this article, the authors used interpolation methods to prove a new version of the limiting case of the Sobolev embedding theorem, where instead of requiring the functions and their derivatives to be in Ln/k, they are required to fit in a rearrangement invariant space A which belongs to a certain class of spaces close to Ln /k.
Abstract: We use interpolation methods to prove a new version of the limiting case of the Sobolev embedding theorem, which includes the result of Hansson and Brezis-Wainger for W
n
k/k
as a special case. We deal with generalized Sobolev spaces W
A
k
, where instead of requiring the functions and their derivatives to be in Ln/k, they are required to be in a rearrangement invariant space A which belongs to a certain class of spaces “close” to Ln/k. We also show that the embeddings given by our theorem are optimal, i.e., the target spaces into which the above Sobolev spaces are shown to embed cannot be replaced by smaller rearrangement invariant spaces. This slightly sharpens and generalizes an, earlier optimality result obtained by Hansson with respect to the Riesz potential operator.
61 citations
TL;DR: In this paper, a generalization of entire functions of spherical exponential type on stratified groups is considered and an analog of the Paley-Wiener theorem is given, where every spectral entire function on a stratified group is uniquely determined by its values on some discrete subgroups.
Abstract: We consider a generalization of entire functions of spherical exponential type on stratified groups. An analog of the Paley-Wiener theorem is given. We also show that every spectral entire function on a stratified group is uniquely determined by its values on some discrete subgroups. The main result of the article is reconstruction formula of spectral entire functions from their values on discrete subgroups using Lagrangian splines.
57 citations
TL;DR: In this paper, an interpolation formula for the expectation of functions of infinitely divisible (i.d.) variables is presented and applied to study the association problem for i.d. vectors and to present new covariance expansions and correlation inequalities.
Abstract: We present an interpolation formula for the expectation of functions of infinitely divisible (i.d.) variables. This is then applied to study the association problem for i.d. vectors and to present new covariance expansions and correlation inequalities.
55 citations
Journal Article•
54 citations
TL;DR: In this article, it was shown that the Wigner distribution of an f ∈ L2(ℝn) cannot be supported by a set of finite measure in ℝ2n unless f = 0.
Abstract: In this note we prove that the Wigner distribution of an f ∈ L2(ℝn) cannot be supported by a set of finite measure in ℝ2n unless f=0. We prove a corresponding statement for cross-ambiguity functions. As a strengthening of the conjecture we show that for an f ∈ L2(ℝn) its Wigner distribution has a support of measure 0 or ∞ in any half-space of ℝ2n.
53 citations
TL;DR: In this article, it was shown that there are subsets K of ℝn such that the single function ψ, defined by a function φ defined by φ = φ K, is an orthonormal wavelet for L 2 (ℝ n).
Abstract: Under very minimal regularity assumptions, it can be shown that 2n−1 functions are needed to generate an orthonormal wavelet basis for L2(ℝn). In a recent paper by Dai et al. it is shown, by abstract means, that there exist subsets K of ℝn such that the single function ψ, defined by
$$\hat \psi = \chi K$$
, is an orthonormal wavelet for L2(ℝn). Here we provide methods for construucting explicit examples of these sets. Moreover, we demonstrate that these wavelets do not behave like their one-dimensional couterparts.
48 citations
TL;DR: The generalized Calderon reproducing formula involving wavelet measure is established for functions f ∈ Lp(ℝn) in this article, which gives rise to the continuous decomposition of f into wavelets, and enables one to obtain inversion formulae for generalized windowed X-ray transforms, the Radon transform, and k-plane transforms.
Abstract: The generalized Calderon reproducing formula involving “wavelet measure” is established for functions f ∈ Lp(ℝn). The special choice of the wavelet measure in the reproducing formula gives rise to the continuous decomposition of f into wavelets, and enables one to obtain inversion formulae for generalized windowed X-ray transforms, the Radon transform, and k-plane transforms. The admissibility conditions for the wavelet measure μ are presented in terms of μ itself and in terms of the Fourier transform of μ.
TL;DR: In this article, it was shown that the Cauchy initial value problem for the Schrodinger equation of a free particle satisfies the following functional inequality, which is derived from a general duality relation of theta-function type.
Abstract: Let
$$h(t,x): = p.v. \sum\limits_{n \in Z\backslash \left| 0 \right|} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} = \mathop {\lim }\limits_{N \to \infty } \sum\limits_{0< \left| n \right| \leqslant N} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} $$
(\((i = \sqrt { - 1;} t,x\)-real variables). It is proved that in the rectangle\(D: = \left\{ {(t,x):0< t< 1,\left| x \right| \leqslant \frac{1}{2}} \right\}\), the function h satisfies the followingfunctional inequality:
$$\left| {h(t,x)} \right| \leqslant \sqrt t \left| {h\left( {\frac{1}{t},\frac{x}{t}} \right)} \right| + c,$$
where c is an absolute positive constant. Iterations of this relation provide another, more elementary, proof of the known global boundedness result
$$\left\| {h; L^\infty (E^2 )} \right\| : = ess sup \left| {h(t,x)} \right|< \infty .$$
The above functional inequality is derived from a general duality relation, of theta-function type, for solutions of the Cauchy initial value problem for Schrodinger equation of a free particle.
TL;DR: In this article, the BMO norm of the logarithm of a weight is controlled by the square root of its A∞ bound, which is the same as in this paper.
Abstract: Sharp inequalities between weight bounds (from the doubling, Ap, and reverse Holder conditions) and the BMO norm are obtained when the former are near their optimal values. In particular, the BMO norm of the logarithm of a weight is controlled by the square root of the logarithm of its A∞ bound. These estimates lead to a systematic development of asymptotically sharp higher integrability results for reverse Holder weights and extend Coifman and Fefferman's formulation of the A∞ condition as an equivalence relation on doubling measures to the setting in which all bounds become optimal over small scales.
TL;DR: In this paper, the existence theory for the inhomogeneous refinement equation with two time scales was developed, where a (k) is a finite sequence and F is a compactly supported distribution on ℝ.
Abstract: Equations with two time scales (refinement equations or dilation equations) are central to wavelet theory. Several applications also include an inhomogeneous forcing term F(t). We develop here a part of the existence theory for the inhomogeneous refinement equation
$$\phi (t) = \sum\limits_{k \in \mathbb{Z}} {a(k)\phi (2t - k) + F(t)}$$
where a (k) is a finite sequence and F is a compactly supported distribution on ℝ.
TL;DR: In this paper, it was shown that having a quasi-metric on a given set X is essentially equivalent to having a family of subsets S(x, r) of X for which y∈S(x and r) implies both S(y, r), r)⊂S (x, Kr) and S(k, r))⊆S(y and r), for some constant K.
Abstract: We prove that having a quasi-metric on a given set X is essentially equivalent to have a family of subsets S(x, r) of X for which y∈S(x, r) implies both S(y, r)⊂S(x, Kr) and S(x, r)⊂S(y, Kr) for some constant K. As an application, starting from the Monge-Ampere setting introduced in [3], we get a space of homogeneous type modeling the real analysis for such an equation.
TL;DR: In this article, it was shown that for certain totally disconnected fractals, function spaces may be characterized by means of the magnitude of the wavelet coefficients of the functions, and that wavelets on self-similar fractals are introduced.
Abstract: Wavelets on self-similar fractals are introduced. It is shown that for certain totally disconnected fractals, function spaces may be characterized by means of the magnitude of the wavelet coefficients of the functions.
TL;DR: In this paper, the fast Fourier transform algorithm was extended to the computation of Fourier transforms on compact Lie groups, and the basic technique uses factorization of group elements and Gel'fand-Tsetlin bases to simplify the computations.
Abstract: This article genralizes the fast Fourier transform algorithm to the computation of Fourier transforms on compact Lie groups. The basic technique uses factorization of group elements and Gel'fand-Tsetlin bases to simplify the computations, and may be extended to treat the computation of Fourier transforms of finitely supported distributions on the group. Similar transforms may be defined on homogeneous spaces; in that case we show how special function properties of spherical functions lead to more efficient algorithms. These results may all be viewed as generalizations of the fast Fourier transform algorithms on the circle, and of recent results about Fourier transforms on finite groups.
[...]
TL;DR: In this article, it was shown that for many (and perhaps all) pairs E, F, of wavelet sets, the corresponding MSF wavelets can be connected by a continuous path in L 2(ℝ) of wavelets for which the Fourier transform has support contained in E ∪ F.
Abstract: It is proved that associated with every wavelet set is a closely related “regularized” wavelet set which has very nice properties. Then it is shown that for many (and perhaps all) pairs E, F, of wavelet sets, the corresponding MSF wavelets can be connected by a continuous path in L2(ℝ) of MSF wavelets for which the Fourier transform has support contained in E ∪ F. Our technique applies, in particular, to the Shannon and Journe wavelet sets.
TL;DR: In this paper, it was shown that every frame for a Hilbert space H can be represented as a linear combination of two orthonormal bases if and only if it is a Riesz basis.
Abstract: We show that every frame for a Hilbert space H can be written as a (multiple of a) sum of three orthonormal bases for H. We next show that this result is best possible by including a result of Kalton: A frame can be represented as a linear combination of two orthonormal bases if and only if it is a Riesz basis. We further show that every frame can be written as a (multiple of a) sum of two tight frames with frame bounds one or a sum of an orthonormal basis and a Riesz basis for H. Finally, every frame can be written as a (multiple of a) average of two orthonormal bases for a larger Hilbert space.
TL;DR: In this article, the authors investigated the asymptotic behavior of wavelets and Bernoulli convolutions using the variational principle and the pressure function, and the results were applied to study the regulatiry of wavelet and wavelet convolutions.
Abstract: Let 0≤g be a dyadic Holder continuous function with period 1 and g(0)=1, and let\(G(x) = \prod
olimits_{n = 0}^\infty {g(x/{\text{2}}^n )} \). In this article we investigate the asymptotic behavior of\(\smallint _0^{\rm T} \left| {G(x)} \right|^q dx\) and\(\frac{1}{n}\sum
olimits_{k = 0}^n {\log g(2^k x)} \) using the dynamical system techniques: the pressure function and the variational principle. An algorithm to calculate the pressure is presented. The results are applied to study the regulatiry of wavelets and Bernoulli convolutions.
TL;DR: In this article, Paley-Wiener theorems for the modified multidimensional Mellin and inverse Mellin transforms are established and the supports of functions are described in terms of their modified Mellin transform without passing to the complexification.
Abstract: New type Paley-Wiener theorems for the modified multidimensional Mellin and inverse Mellin transforms are established. The supports of functions are described in terms of their modified Mellin (or inverse Mellin) transform without passing to the complexification.
TL;DR: In this paper, the deconvolution (or Bezout) equation (0.1) is solved using sampling of bandlimited functions, which can be easily implemented numerically.
Abstract: Explicit, compactly supported solutions, {vi, ϕ}
i=1
m
, to the deconvolution (or Bezout) equation
(0.1)
$$\sum\limits_{i = 1}^m {\mu _i *}
u _{i,\Phi } = \Phi$$
are computed where ϕ is a given function in C
c
∞
(Rd), and
$$\mu _i = \chi _{[ - r_i ,r_j ]^d }$$
, i=1, ..., m for some set of positive numbers {ri}
i=1
m
such that ri/rj is poorly approximated by rationals whenever i ≠ j. The novelty of the solution technique is that it uses new results in the theory of sampling of bandlimited functions detailed in [13] to provide simple Fourier series representations for the solutions, {vi, ϕ}
i=1
m
, which can be easily implemented numerically. Several examples illustrating the use of sampling for solutions to variants of (0.1) are given, as well as some numerical simulations.
TL;DR: This paper introduces concepts of Gabor-Gram matrices and investigates their structural properties, and proposes iterative methods to compute the Gabor coefficients.
Abstract: The fundamental problem ofdiscrete Gabor transforms is to compute a set ofGabor coefficients in efficient ways. Recent study on the subject is an indirect approach: in order to compute the Gabor coefficients, one needs to find an auxiliary bi-orthogonal window function γ.
TL;DR: In this article, some restriction theorems for flat homogeneous surfaces of codimension greater than one were proved for the case of flat surfaces with codimensions greater than 1.
Abstract: We prove some restriction theorems for flat homogeneous surfaces of codimension greater than one.
TL;DR: In this article, the boundary behavior of harmonic functions in a domain whose boundary is locally given by a graph of a Holder continuous function is analyzed, and a non-probabilistic proof of a Harnack-type principle is given.
Abstract: We analyze the boundary behavior of harmonic functions in a domain whose boundary is locally given by a graph of a Holder continuous function. In particular we give a non-probabilistic proof of a Harnack-type principle, due to Banuelos et al. and study some properties of the harmonic measure.
TL;DR: In this paper, the authors studied the well-posedness of the initial value problem of the semilinear heat equation for k = 2 with μ 0 ǫ hs(ℝn) with initial data in Ω(n, r, p, s, and u0).
Abstract: This article mainly consists of two parts. In the first part the initial value problem (IVP) of the semilinear heat equation
$$\begin{gathered} \partial _t u - \Delta u = \left| u \right|^{k - 1} u, on \mathbb{R}^n x(0,\infty ), k \geqslant 2 \hfill \\ u(x,0) = u_0 (x), x \in \mathbb{R}^n \hfill \\ \end{gathered} $$
with initial data in\(\dot L_{r,p} \) is studied. We prove the well-posedness when
$$1< p< \infty , \frac{2}{{k(k - 1)}}< \frac{n}{p} \leqslant \frac{2}{{k - 1}}, and r =< \frac{n}{p} - \frac{2}{{k - 1}}( \leqslant 0)$$
and construct non-unique solutions for
$$1< p< \frac{{n(k - 1)}}{2}< k + 1, and r< \frac{n}{p} - \frac{2}{{k - 1}}.$$
In the second part the well-posedness of the avove IVP for k=2 with μ0ɛHs(ℝn) is proved if
$$ - 1< s, for n = 1, \frac{n}{2} - 2< s, for n \geqslant 2.$$
and this result is then extended for more general nonlinear terms and initial data. By taking special values of r, p, s, and u0, these well-posedness results reduce to some of those previously obtained by other authors [4, 14].
TL;DR: In this paper, a Calderon reproducing formula for a continuous wavelet transform associated with a class of singular differential operators on the half line is presented, and the authors apply this result to derive a new inversion formula for the generalized Abel transform.
Abstract: We prove a Calderon reproducing formula for a continuous wavelet transform associated with a class of singular differential operators on the half line. We apply this result to derive a new inversion formula for the generalized Abel transform.
TL;DR: Gibbs' phenomenon occurs for most orthogonal wavelet expansions as mentioned in this paper, and it is also shown to occur with many wavelet interpolating series, and a characterization is given.
Abstract: Gibbs' phenomenon occurs for most orthogonal wavelet expansions. It is also shown to occur with many wavelet interpolating series, and a characterization is given. By introducing modifications in such a series, it can be avoided. However, some series that exhibit Gibbs' phenomenon for orthogonal series do not for the associated sampling series.
TL;DR: In this paper, the constructions of biorthogonal basis of compactly supported wavelets in Sobolev spaces of integer order were studied, using techniques of [1] and [2].
Abstract: This article is concerned with constructions of biorthogonal basis of compactly supported wavelets in Sobolev spaces of integer order. Using techniques of [1] and [2], the results presented here generalize to Sobolev spaces some constructions of Cohen et al. [7] and Chui and Wang [5] established in L2(ℝ).
TL;DR: In this article, the authors give necessary conditions for subsets of the unit circle to give collections of rectangles (by means of orientations) which differentiate Lp-functions or give Hardy-Littlewood type maximal functions which are bounded on Lp, p>1.
Abstract: In this article we give some new necessary conditions for subsets of the unit circle to give collections of rectangles (by means of orientations) which differentiate Lp-functions or give Hardy-Littlewood type maximal functions which are bounded on Lp, p>1. This is done by proving that a well-known method, the construction of a Perron Tree, can be applied to a larger collection of subsets of the unit circle than was earlier known. As applications, we prove a partial converse of a well-known result of Nagel et al. [6] regarding boundedness of maximal functions with respect to rectangles of lacunary directions, and prove a result regarding the cardinality of subsets of arithmetic progressions in sets of the type described above.