We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions (which we will call q-chirps for short), giving rise to a parameter space that includes both the time-frequency plane and the time-scale plane as two-dimensional subspaces. The parameter space contains a "time-frequency-scale volume ", and thus encompasses both the short-time Fourier transform (as a slice along the time and frequency axes), and the wavelet transform (as a slice along the time and scale axes). In addition to time, frequency, and scale, there are two other coordinate axes within this transform space: shear-in-time (obtained through convolution with a q-chirp) and shearin -frequency (obtained through multiplication by a q-chirp). Signals in this multidimensional space can be obtained by a new transform which we call the "q-chirplet transform", or simply the "chiplet transform". ...

In this paper we provide several generalizations of inequalities bounding the commutator of two linear operators acting on a Hilbert space which relate to the Heisenberg uncertainty principle and time/frequency analysis of periodic functions. We develop conditions that ensure these inequalities are sharp and apply our results to concrete examples of importance in the literature.

In this paper we investigate quadrature rules for functions on compact Lie groups and sections of homogeneous vector bundles associated with these groups. First a general notion of bandlimitedness is introduced which generalizes the usual notion on the torus or translation groups. We develop a sampling theorem that allows exact computation of the Fourier expansion of a band-limited function or section from sample values and quantifies the error in the expansion when the function or section is not band-limited. We then construct specific finitely supported distributions on the classical groups which have nice error properties and can also be used to develop efficient algorithms for the computation of Fourier transforms on these groups. Contents 1 Introduction 1 2 Sampling of Functions 3 2.1 An abstract framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Sampling of Functions on a Compact Lie Group . . . . . . . . . . . . . . . . . . . 5 2.2.1 Notation a...

Data produced by a reproducible source contains redundant information which allows seismic inversion to simultaneously determine the high-frequency fluctuation in the P-wave velocity (or reflectivity) as well as the input energy source. The seismogram model is the plane-wave convolutional model derived from the constant density, variable sound velocity acoustic wave equation. The first step is to analyze this linearized model when the background velocity is constant. Then perturbations in the seismic data stably determine corresponding perturbations in the source and reflectivity. The stability of this determination improves as the slowness aperture over which the data is defined increases. Further, the normal operator for the convolutional seismogram model is continuous with respect to velocity. Thus the stability result for constant background velocities may be extended to more realistic background velocity models which vary slowly and smoothly with depth. The theory above is illustr...

We propose a new method to analyze and represent data recorded on a domain of gen-eral shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonal-ize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neu-mann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the compu-tation. We also show that our method is better suited for small sample data than the Karhunen-Loève Transform/Principal Component Analysis. In fact, our eigenfunctions depend only on the shape of the domain, not the statistics of the data. As a further ap-plication, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain. Key words: Laplacian eigenfunctions, boundary conditions, Green’s function, spectral decomposition, Karhunen-Loève transform, principal component analysis, heat equation

Abstract. The classical (scalar-valued) theory of spherical functions, put forward by Cartan and others, unifies under one roof a number of examples that were very well-known before the theory was formulated. These examples include special functions such as like Jacobi polynomials, Bessel functions, Laguerre polynomials, Hermite polynomials, Legendre functions, which had been workhorses in many areas of mathematical physics before the appearance of a unifying theory. These and other functions have found interesting applications in signal processing, including specific areas such as medical imaging. The theory of matrix-valued spherical functions is a natural extension of the well-known scalar-valued theory. Its historical development, however, is different: in this case the theory has gone ahead of the examples. The purpose of this article is to point to some examples and to interest readers in this new aspect in the world of special functions. We close with a remark connecting the functions described here with the theory of matrix-valued orthogonal polynomials. 1.

. In this survey we review recent developments concerning the ecient iterative regularization of image reconstruction problems in atmospheric imaging. We present a number of preconditioners for the minimization of the corresponding Tikhonov functional, and discuss the alternative of terminating the iteration early, rather than adding a stabilizing term in the Tikhonov functional. The methods are examplied for a (synthetic) model problem. 1 Introduction Atmospheric turbulences are the reason for severe problems in ground based astronomical imaging. On the passage through the atmosphere, light waves are scattered because of temperature uctuations both in space and time, which lead to strong aberrations of astronomical images taken by a telescope on the surface of the Earth. In principle, if a sophisticated model of the scattering process is available, the true image can be reconstructed from the photo by solving the associated inverse problem. Such models, however, are very di...

Applying the fractional Fourier transform and the Wigner distribution on a signal in a cascade fashion is equivalent with a rotation of the time and frequency parameters of the Wigner distribution. This report presents a formula for all unitary operators that are related to energy preserving transformations on the parameters of the Wigner distribution by means of such a cascade of operators. Furthermore, such operators are used to solve certain type of energy localization problems via the Weyl correspondence.

Satellites mapping out the spatial variations of the gravitational or magnetic fields of the Earth or other planets ideally fly on polar orbits, uniformly covering the entire globe. Thus potential fields on the sphere are usually expressed in spherical harmonics, basis functions with global support. For various, especially engineering, reasons, however, inclined orbits are favorable. These leave a “polar gap”: an antipodal pair of axisymmetric polar caps, typically less than 10 ◦ in diameter, without any data coverage. Estimation of spherical harmonic field coefficients from an incompletely sampled sphere is prone to error, since the spherical harmonics are not orthogonal over the partial domain of the cut sphere. The historically somewhat neglected geodetic polar gap problem has been revived by, among others, Sneeuw & van Gelderen (1997), and recently,

Abstract. In this survey, we present various forms of the uncertainty principle (Hardy, Heisenberg, Benedicks). We further give a new interpretation of the uncertainty principles as a statement about the time-frequency localization of elements of an orthonormal basis, which improves previous unpublished results of H. Shapiro. Finally, we show that Benedicks ’ result implies that solutions of the Shrödinger equation have some (appearently unnoticed) energy dissipation property. 1.