Coset decomposition of positively self-conjugate inverse submonoids of polycyclic monoids

We obtain several new results about the coding of a monoid by an appropriate submonoid of a polycyclic monoid. In particular, we characterize groups, periodic monoids, and right cancellative monoids in terms of the coset decomposition of positively self-conjugate inverse submonoids of polycyclic monoids.     

Coset Decomposition of Positively Self-Conjugate Inverse Submonoids of Polycyclic Monoids

Abstract. We obtain several new results about the coding of a monoid by an appropriate submonoid of a polycyclic monoid. In particular, we characterize groups, periodic monoids, and right cancellative monoids in terms of the coset decomposition of positively self-conjugate inverse submonoids of polycyclic monoids.     

Asymptotic analysis of quantum dynamics in crystals: the Bloch-Wigner transform, Bloch dynamics and Berry phase

We study the semi-classical limit of the Schro¨dinger equation in a crystal in the presence of an external potential and magnetic field. We first introduce the Bloch-Wigner transform and derive the asymptotic equations governing this transform in the semi-classical setting. For the second part, we focus on the appearance of the Berry curvature terms in the asymptotic equations. These terms play a crucial role in many important physical phenomena such as the quantum Hall effect. We give a simple derivation of these terms in different settings using asymptotic analysis.     

Wavelet transform associated with linear canonical Hankel transform

The main goal of this paper is to study about the continuous as well as discrete wavelet transform in terms of linear canonical Hankel transform (LCH‐transform) and discuss some of its basic properties. Parseval's relation and reconstruction formula of continuous linear canonical Hankel wavelet transform (CLCH‐wavelet transform) is obtained. Moreover, semidiscrete and discrete LCH‐wavelet transform are also discussed.     

Multidimensional Cooley–Tukey Algorithms Revisited

The representation theory of Abelian groups is used to obtain an algebraic divide-and-conquer algorithm for computing the finite Fourier transform. The algorithm computes the Fourier transform of a finite Abelian group in terms of the Fourier transforms of an arbitrary subgroup and its quotient. From this algebraic algorithm a procedure is derived for obtaining concrete factorizations of the Fourier transform matrix in terms of smaller Fourier transform matrices, diagonal multiplications, and permutations. For cyclic groups this gives as special cases the Cooley–Tukey and Good–Thomas algorithms. For groups with several generators, the procedure gives a variety of multidimensional Cooley–Tukey type algorithms. This method of designing multidimensional fast Fourier transform algorithms gives different data flow patterns from the standard “row–column” approaches. We present some experimental evidence that suggests that in hierarchical memory environments these data flows are more efficient.     

Direct form seminorms arising in the theory of interpolation by Hankel translates of a basis function

Certain spaces of functions which arise in the process of interpolation by Hankel translates of a basis function, as developed by the authors elsewhere, are defined with respect to a seminorm which is given in terms of the Hankel transform of each function. This kind of seminorm is called an indirect one. Here we discuss essentially two cases in which the seminorm can be rewritten in direct form, that is, in terms of the function itself rather than its Hankel transform. This is expected to lead to better estimates of the interpolation error.     

A characterization of symmetric tube domains by convexity of Cayley transform images

In this paper, we show that a homogeneous tube domain is symmetric if and only if its Cayley transform image as well as the dual Cayley transform image of the dual tube domain is convex. In this case, the parameters of these Cayley transforms reduce to specific ones, so that they are essentially the usual Cayley transforms defined in terms of Jordan algebra structure.     

Growth properties of Fourier transforms via moduli of continuity

We obtain new inequalities for the Fourier transform, both on Euclidean space, and on non-compact, rank one symmetric spaces. In both cases these are expressed as a gauge on the size of the transform in terms of a suitable integral modulus of continuity of the function. In all settings, the results present a natural corollary: a quantitative form of the Riemann-Lebesgue lemma. A prototype is given in one-dimensional Fourier analysis.     

Two-dimensional quaternion wavelet transform

In this paper we introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several examples of the CQWT. As an application we derive a Heisenberg type uncertainty principle for these extended wavelets.     

Stabilization and error bounds for the inverse laplace transform

We present some theorems on the stabilization of the inverse Laplace transform. We assume the Laplace transform is measured at N points to within some error ?. We prove error bounds for the inverse Laplace transform in terms of ? and N, under suitable a-priori constraints. This is achieved by proving parallel stabilization results for a related Hausdorff moment problem.     

Exact equations for smoothed Wigner transforms and homogenization of wave propagation

The Wigner transform (WT) has been extensively used in the formulation of phase-space models for a variety of wave propagation problems including high-frequency limits, nonlinear and random waves. It is well known that the WT features counterintuitive ‘interference terms,’ which often make computation impractical. In this connection, we propose the use of the smoothed Wigner transform (SWT), and derive new, exact equations for it, covering a broad class of wave propagation problems. Equations for spectrograms are included as a special case. The ‘taming’ of the interference terms by the SWT is illustrated, and an asymptotic model for the Schrödinger equation is constructed and numerically verified.     

Sparse directional image representations using the discrete shearlet transform

In spite of their remarkable success in signal processing applications, it is now widely acknowledged that traditional wavelets are not very effective in dealing multidimensional signals containing distributed discontinuities such as edges. To overcome this limitation, one has to use basis elements with much higher directional sensitivity and of various shapes, to be able to capture the intrinsic geometrical features of multidimensional phenomena. This paper introduces a new discrete multiscale directional representation called the discrete shearlet transform. This approach, which is based on the shearlet transform, combines the power of multiscale methods with a unique ability to capture the geometry of multidimensional data and is optimally efficient in representing images containing edges. We describe two different methods of implementing the shearlet transform. The numerical experiments presented in this paper demonstrate that the discrete shearlet transform is very competitive in denoising applications both in terms of performance and computational efficiency.     

A Support Theorem for the Complex Radon Transform of Distributions

The properties of the complex Radon transform of compactly supported distributions are considered. For such distributions, we prove a support theorem allowing us to describe the support of the distribution in terms of the support of its Radon transform.     

The Fractional Clifford-Fourier Transform

In this paper, a fractional version of the Clifford-Fourier transform is introduced, depending on two numerical parameters. A series expansion for the kernel of the resulting integral transform is derived. In the case of even dimension, also an explicit expression for the kernel in terms of Bessel functions is obtained. Finally, the analytic properties of this new integral transform are studied in detail.     

Quadrature formulas for the Laplace and Mellin transforms

A discrete Laplace transform and its inversion formula are obtained by using a quadrature of the integral Fourier transform which is given in terms of Hermite polynomials and its zeros. This approach yields a convergent discrete formula for the two-sided Laplace transform if the function to be transformed falls off rapidly to zero and satisfies given conditions of integrability, achieving convergence also for singular functions. The inversion formula becomes a quadrature formula for the Bromwich integral. The use of asymptotic formulae yields an algorithm to compute the discrete Laplace transform by using only exponentials.     

On topological Radon transformations

We construct a functor, which we call the topological Radon transform, from a category of complex algebraic varieties with morphisms given by divergent diagrams, to constructible functions. The topological Radon transform is thus the composition of a pull-back and a push-forward of constructible functions. We show that the Chern-Schwartz-MacPherson transformation makes the topological Radon transform of constructible functions compatible with a certain homological Verdier-Radon transform. We use this set-up to prove, given a projective variety X, a formula for the Chern-Mather class of the dual variety in terms of that of X.     

Inversion and jump formulae for variation diminishing transforms

Summary This is a study of the values of the determining function φ as well as its derivatives and gaps in terms of the generating function f and its derivatives where f is the convolution transform of φ with variation diminishing kernel. Entrata in Redazione il 26 aprile 1968.     

Asymptotics of the Fourier sine transform of a function of bounded variation

For the asymptotic formula for the Fourier sine transform of a function of bounded variation, we find a new proof entirely within the framework of the theory of Hardy spaces, primarily with the use of the Hardy inequality. We show that, for a function of bounded variation whose derivative lies in the Hardy space, every aspect of the behavior of its Fourier transform can somehow be expressed in terms of the Hilbert transform of the derivative.     

Numerical inversion of Mellin transforms

A method is presented for the numerical inversion of Mellin transforms in which the inverse is obtained as an expansion in terms of Laguerre polynomials. The coefficients of this expansion are obtained as linear combinations of values of the transformed function or, equivalently, in terms of forward differences of this function. Thus, the Mellin transform of the series can be written as a forward interpolation series. Consequently the error of the numerical inversion procedure can be estimated. The practical advantage of the method is that values are needed for real arguments only.