Discrete Wiener¿Lee transforms

The use of Wiener–Lee transforms to construct one of the frequency characteristics, magnitude or phase of a network function, when the other characteristic is given graphically, is indicated. This application is useful in finding a realisable network function whose magnitude or phase curve is given. A discrete version of the transform is presented, so that a digital computer can be employed for the computation.

Revisiting Riesz transforms on Heisenberg groups

We characterise higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimension-free bounds under some assumptions on the multipliers. Using transference theorems, we deduce boundedness theorems for Riesz transforms on the reduced Heisenberg group and hence also for the Riesz transforms associated to multiple Hermite and Laguerre expansions.

Compressive sensing for music signals: comparison of transforms with coherent dictionaries

Compressive Sensing (CS) is a new sensing paradigm which permits sampling of a signal at its intrinsic information rate which could be much lower than Nyquist rate, while guaranteeing good quality reconstruction for signals sparse in a linear transform domain. We explore the application of CS formulation to music signals. Since music signals comprise of both tonal and transient nature, we examine several transforms such as discrete cosine transform (DCT), discrete wavelet transform (DWT), Fourier basis and also non-orthogonal warped transforms to explore the effectiveness of CS theory and the reconstruction algorithms. We show that for a given sparsity level, DCT, overcomplete, and warped Fourier dictionaries result in better reconstruction, and warped Fourier dictionary gives perceptually better reconstruction. “MUSHRA” test results show that a moderate quality reconstruction is possible with about half the Nyquist sampling.

Riesz transforms and multipliers for the Grushin operator

We show that Riesz transforms associated to the Grushin operator G = -Delta - |x|(2 similar to) (t) (2) are bounded on L (p) (a''e (n+1)). We also establish an analogue of the Hormander-Mihlin Multiplier Theorem and study Bochner-Riesz means associated to the Grushin operator. The main tools used are Littlewood-Paley theory and an operator-valued Fourier multiplier theorem due to L. Weis.

L-p-Integrability, Dimensions of Supports of Fourier Transforms and Applications

It is proved that there does not exist any non zero function in with if its Fourier transform is supported by a set of finite packing -measure where . It is shown that the assertion fails for . The result is applied to prove L-p Wiener Tauberian theorems for R-n and M(2).

DIMENSION FREE BOUNDEDNESS OF RIESZ TRANSFORMS FOR THE GRUSHIN OPERATOR

Let G = -Delta(xi) - vertical bar xi vertical bar(2) partial derivative(2)/partial derivative eta(2) be the Grushin operator on R-n x R. We prove that the Riesz transforms associated to this operator are bounded on L-p(Rn+1), 1 < p < infinity, and their norms are independent of dimension n.

Mixed norm estimates for the Riesz transforms on the Heisenberg group

In this paper we prove weighted mixed norm estimates for Riesz transforms on the Heisenberg group and Riesz transforms associated to the special Hermite operator. From these results vector-valued inequalities for sequences of Riesz transforms associated to generalised Grushin operators and Laguerre operators are deduced.

MIXED NORM ESTIMATES FOR THE RIESZ TRANSFORMS ON SU (2)

In this paper we prove mixed norm estimates for Riesz transforms on the group SU(2). From these results vector valued inequalities for sequences of Riesz transforms associated to Jacobi differential operators of different types are deduced.

Dynamic behavior of a cylindrical crack in a functionally graded interlayer under torsional loading

In this paper the problem of a cylindrical crack located in a functionally graded material (FGM) interlayer between two coaxial elastic dissimilar homogeneous cylinders and subjected to a torsional impact loading is considered. The shear modulus and the mass density of the FGM interlayer are assumed to vary continuously between those of the two coaxial cylinders. This mixed boundary value problem is first reduced to a singular integral equation with a Cauchy type kernel in the Laplace domain by applying Laplace and Fourier integral transforms. The singular integral equation is then solved numerically and the dynamic stress intensity factor (DSIF) is also obtained by a numerical Laplace inversion technique. The DSIF is found to rise rapidly to a peak and then reduce and tend to the static value almost without oscillation. The influences of the crack location, the FGM interlayer thickness and the relative magnitudes of the adjoining material properties are examined. It is found among others that, by increasing the FGM gradient, the DSIF can be greatly reduced.

Torsional impact response of a penny-shaped interface crack in bonded materials with a graded material interlayer

In this paper, the dynamic response of a penny-shaped interface crack in bonded dissimilar homogeneous half-spaces is studied. It is assumed that the two materials are bonded together with such a inhomogeneous interlayer that makes the elastic modulus in the direction perpendicular to the crack surface is continuous throughout the space. The crack surfaces art assumed to be subjected to torsional impact loading. Laplace and Hankel integral transforms are applied combining with a dislocation density,function to reduce the mixed boundary value problem into a singular integral equation with a generalized Cauchy kernel in Laplace domain. By solving the singular integral equation numerically, and using a numerical Laplace inversion technique, the dynamic stress intensity factors art obtained. The influences of material properties and interlayer thickness on the dynamic stress intensity factor are investigated.