Wave packet transform and fractional wave packet transform of rapidly decreasing functions

The continuity of wave packet transform and inverse wave packet transform is proved in the suitable Schwartz space and extended to its corresponding dual space of tempered distribution. The consistency, linearity and continuity of the transform with respect to [Formula: see text] topology are proved in this distribution space. Further, the continuity of the fractional wave packet transform and its inverse in the above space is proved. The examples of generalized fractional wave packet transform of certain distributions are given.

Continuous Wavelet Transform of Schwartz Tempered Distributions in S′ ( R n )

In this paper, we define a continuous wavelet transform of a Schwartz tempered distribution f ∈ S ′ ( R n ) with wavelet kernel ψ ∈ S ( R n ) and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of S ′ ( R n ) . It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution.

On the Fourier Transformability of Strongly Almost Periodic Measures

In this paper we characterize the Fourier transformability of strongly almost periodic measures in terms of an integrability condition for their Fourier–Bohr series. We also provide a necessary and sufficient condition for a strongly almost periodic measure to be the Fourier transform of a measure. We discuss the Fourier transformability of a measure on $\mathbb{R}^{d}$ in terms of its Fourier transform as a tempered distribution. We conclude by looking at a large class of such measures coming from the cut and project formalism.

Continuous Wavelet Transform of Schwartz Tempered Distributions in $S'(\mathbb R^n)$

In this paper we define a continuous wavelet transform of a Schwartz tempered distribution $f \in S^{'}(\mathbb R^n)$ with wavelet kernel $\psi \in S(\mathbb R^n)$ and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of $S^{'}(\mathbb R^n)$. It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution.

Curvelet transform on tempered distributions

The curvelet transform of a tempered distribution is defined as an infinitely differentiable function of (a, b, θ) with a polynomial growth in b. An inversion formula of the curvelet transform on tempered distributions is also obtained.

A gradient-projective basis of compactly supported wavelets in dimension n > 1

A given set W = {W X } of n-variable class C 1 functions is a gradient-projective basis if for every tempered distribution f whose gradient is square-integrable, the sum $\sum\limits_\chi {(\int_{\mathbb{R}^n } {\nabla f \cdot } \nabla W_\chi ^* )} W_\chi $ converges to f with respect to the norm \(\left\| {\nabla ( \cdot )} \right\|_{L^2 (\mathbb{R}^n )} \) . The set is not necessarily an orthonormal set; the orthonormal expansion formula is just an element of the convex set of valid expansions of the given function f over W. We construct a gradient-projective basis W = {W x } of compactly supported class C 2−ɛ functions on ℝn such that

A Note on Directional Wavelet Transform: Distributional Boundary Values and Analytic Wavefront Sets

By using a particular class of directional wavelets (namely, the conical wavelets, which are wavelets strictly supported in a proper convex cone in thek-space of frequencies), in this paper, it is shown that a tempered distribution is obtained as a finite sum of boundary values of analytic functions arising from the complexification of the translational parameter of the wavelet transform. Moreover, we show that for a given distributionf∈𝒮′(ℝn), the continuous wavelet transform offwith respect to a conical wavelet is defined in such a way that the directional wavelet transform offyields a function on phase space whose high-frequency singularities are precisely the elements in the analytic wavefront set off.

On the support of tempered distributions

We show that if the summability means in the Fourier inversion formula for a tempered distribution f ∈ S′(ℝn) converge to zero pointwise in an open set Ω, and if those means are locally bounded in L1(Ω), then Ω ⊂ ℝn\supp f. We prove this for several summability procedures, in particular for Abel summability, Cesàro summability and Gauss-Weierstrass summability.

The support of tempered distributions

We identify the support of a tempered distribution by evaluation of a sequence of test functions against the Fourier transform of the distribution. This improves previous results by removing the restriction that the distribution's Fourier transform be in $L^1_{loc}$ and be of polynomial growth. We use an apparently new technical lemma that implies that certain bounded approximate identities for $L^1(\R^n)$ are also topological approximate identities for elements of the space $\Sl$ of Schwartz functions.