where Jν(x) is the Bessel function of the first kind of order ν [1], and =z denotes the imaginary part of z. An extensive table of integral transforms involving the Bessel functions in the kernels is collected in [6]. Since the integration in (2) is with respect to the order of the Bessel function, such a pair of integral transforms is called index transform. Details about many other index tran...

A chromophore in which zinc porphyrin donors are linked through their meso positions by ethynyl bridges to a bis(indolinylidenemethyl) squaraine core has been synthesized using Sonogashira coupling. The chromophore exhibits a two-photon absorption spectrum characterized by a peak cross section of 11,000 GM and, more unusually, also exhibits a large cross section of >780 GM over a photon-wavelen...

Let φ : C → C be a bilipschitz map. We prove that if E ⊂ C is compact, and γ(E), α(E) stand for its analytic and continuous analytic capacity respectively, then C−1γ(E) ≤ γ(φ(E)) ≤ Cγ(E) and C−1α(E) ≤ α(φ(E)) ≤ Cα(E), where C depends only on the bilipschitz constant of φ. Further, we show that if μ is a Radon measure on C and the Cauchy transform is bounded on L2(μ), then the Cauchy transform i...

A function φ is refinable (φ ∈ S) if it is in the closed span of {φ(2x−k)}. This set S is not closed in L2(R), and we characterize its closure. A necessary and sufficient condition for a function to be refinable is presented without any information on the refinement mask. The Fourier transform of every f ∈ S \ S vanishes on a set of positive measure. As an example, we show that all functions wi...

We present a new class of wavelet bases—Fresnelets—which is obtained by applying the Fresnel transform operator to a wavelet basis of L2. The thus constructed wavelet family exhibits properties that are particularly useful for analyzing and processing optically generated holograms recorded on CCD-arrays. We first investigate the multiresolution properties (translation, dilation) of the Fresnel ...

Using elementary arguments based on the Fourier transform we prove that for 1 ≤ q < p < ∞ and s ≥ 0 with s > n(1/2 − 1/p), if f ∈ L(R) ∩ Ḣ(R) then f ∈ L(R) and there exists a constant cp,q,s such that ‖f‖Lp ≤ cp,q,s‖f‖ θ Lq,∞‖f‖ 1−θ Ḣs , where 1/p = θ/q+(1− θ)(1/2− s/n). In particular, in R we obtain the generalised Ladyzhenskaya inequality ‖f‖L4 ≤ c‖f‖ 1/2 L2,∞ ‖f‖ 1/2 Ḣ1 . We also show that f...

We present a set of functions in L2([0,∞)) and show it to be a (tight) generalized frame (as presented by G. Kaiser (1994)). The analysis side of the frame operation is called the continuous unified transform. We show that some of the well-known transforms (such as Laplace, Laguerre, Kautz, and Hambo) result by creating different sampling patterns in the transform domain (or, equivalently, choo...

The analysis of transient signals is an attractive framework for many problems in digital signal processing. This paper provides an application of the data adaptive linear decomposition transform (LDT) to detect transients in the sinusoidal signal corrupted by an additive impulse noise. The LDT procedure entails obtaining an interpolation error sequence of half-length of the sinusoidal signal, ...

If /(OG^P) £>!> then f(x)^Lpy and a considerable literature is devoted to studying the relationship of such pairs of "conjugate" functions to the theory of functions analytic in a half-plane. More to the point of the present note is a series of papers studying the Hubert transform along strictly real variable lines ([2, 3 ] ; further bibliography in [2]). Much less is known about f(x) when f(t)...

This study aims to achieve an efficient time-frequency representation of higher-dimensional signals by introducing the notion a non-separable linear canonical wavelet transform in L2(Rn). The preliminary analysis encompasses derivation fundamental properties novel integral including orthogonality relation, inversion formula, and range theorem. To extend scope study, we formulate several uncerta...