Symmetric Convolution Research Articles

Spectral functions of self-adjoint and symmetric multiplication operators inL 2(X, μ)-spaces

Explicit formulas are derived for the spectral function of double multiplication operator containing a multiplicative evolution inL2(X, μ)-space and a convolution-type operator inL2(ℝn)-spaces. Symmetric convolution and multiplication operators are considered inL2(X, μ) andL2(ℝn)-spaces.

Symmetric convolution of asymmetric multidimensional sequences using discrete trigonometric transforms

This paper uses the fact that the discrete Fourier transform diagonalizes a circulant matrix to provide an alternate derivation of the symmetric convolution-multiplication property for discrete trigonometric transforms. Derived in this manner, the symmetric convolution-multiplication property extends easily to multiple dimensions using the notion of block circulant matrices and generalizes to multidimensional asymmetric sequences. The symmetric convolution of multidimensional asymmetric sequences can then be accomplished by taking the product of the trigonometric transforms of the sequences and then applying an inverse trigonometric transform to the result. An example is given of how this theory can be used for applying a two-dimensional (2-D) finite impulse response (FIR) filter with nonlinear phase which models atmospheric turbulence.

Image reconstruction using symmetric convolution and discrete trigonometric transforms

We demonstrate how the symmetric convolution-multiplication property of the discrete trigonometric transforms can be applied to problems in image reconstruction. This property allows for linear filtering of degraded images by means of point-by-point multiplication in the transform domain of trigonometric transforms. Specifically, in the transform domain of a type II discrete cosine transform, there is an asymptotically optimum energy compaction near d.c. for highly correlated images, which has advantages in reconstructing images with high-frequency noise. The symmetric convolution-multiplication property allows for scalar representations in the transform-domain space of discrete trigonometric transforms for linear reconstruction filters such as the Wiener filter. An analysis of the scalar Wiener filter’s performance in the trigonometric transform domain is given.

Zero-One Law for Symmetric Convolution Semigroups of Measures on Groups

Let (μ t ) t>0 be a symmetric weakly continuous semigroup of probability measures on a nonabelien complete separable group G and let v be its Levy measure. The purpose of this paper is to provide a relatively simple proof of the zero-one law for semigroups with the Levy measure satisfying either v(H c) = ∞ or v(H c) = 0.

Digital filtering of images using the discrete sine or cosine transform

The discrete sine and cosine transforms (DSTS and DCTS) are powerful tools for image compression. They would be even more useful if they could be used to perform convolution. As we have recently shown, these transforms do possess a convolution-multiplication property and therefore can be used to perform digital filtering. The convolution is a special type, called symmetric corrvo/ution. This paper reviews the concepts of symmetric convolution and then elaborates on how to use the operation, and thereby DSTS and DCTS, to implement digital filters for images. An indication of how to compute the transforms and what is the complexity of their use is also given.

Symmetric convolution and the discrete sine and cosine transforms

This paper discusses the use of symmetric convolution and the discrete sine and cosine transforms (DSTs and DCTs) for general digital signal processing. The operation of symmetric convolution is a formalized approach to convolving symmetrically extended sequences. The result is the same as that obtained by taking an inverse discrete trigonometric transform (DTT) of the product of the forward DTTs of those two sequences. There are 16 members in the family of DTTs. Each provides a representation for a corresponding distinct type of symmetric-periodic sequence. The author defines symmetric convolution, relates the DSTs and DCTs to symmetric-periodic sequences, and then use these principles to develop simple but powerful convolution-multiplication properties for the entire family of DSTs and DCTs. Symmetric convolution can be used for discrete linear filtering when the filter is symmetric or antisymmetric. The filtering will be efficient because fast algorithms exist for all versions of the DTTs. Conventional linear convolution is possible if one first zero-pad the input data. Symmetric convolution and its fast implementation using DTTs are now an alternative to circular convolution and the DFT. >

Invariant measures of symmetric Lévy processes

If π = { π t : t > 0 } \pi = \{ {\pi _t}:t > 0\} is a symmetric convolution semigroup with the Lévy exponent ϕ \phi , then supp π t {\pi _t} , is a group determined by ϕ \phi , and π \pi has a unique Radon invariant measure if and only if ϕ \phi has a unique zero at 0 0 .

Systolic pipeline architectures for symmetric convolutions

Symmetries in the coefficients of common types of finite-impulse response (FIR) digital filters can be utilized to enable systolic pipeline realizations with considerable reduction in the number of multiplier stages. To obtain a concise description of the interconnections of these pipeline structures, a recursive state variable representation is presently developed. The representation is then utilized to examine the effects of symmetry on pipeline structures for discrete convolutions in one and two dimensions. For common types of one-dimensional filters, such as bandpass, phase-shift (Hilbert transform), and edge detector (differentiator) types, the number of multipliers may be approximately halved using odd/even symmetry. For common types of two-dimensional filters, such as bandpass, rho filters (tomographics), and edge detector (Laplacian) types, the number of multipliers can be reduced by approximately a factor of eight.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

About polynomials related to multiphotonic bremsstrahlung effects

The symmetric convolution of two Hermite functions appears as transition amplitude in the multiphotonic bremsstrahlung effect analysed by quantum oscillator models. The authors identify these transition amplitudes in terms of Laguerre polynomials and give new recurrence relations and a differential equation for these physical quantities.

Finite symmetric convolution operators and singular symmetric differential operators

In this note, we prove a generalization of a theorem of Morrison ( Notices Amer. Math. Soc. 1962, 119) which states that the only real finite symmetric convolution integral operator K in L 2[−1, 1] with kernel ϱ analytic about the origin, which admits a second order, singular symmetric differential operator in its commutator is (aside from a trivial scaling) given by Kφ( x) = ∝ −1 +1 ϱ( x − y) φ( y) dy, φ ϵ L 2[−1, 1], with ϱ(x) = Ω 2 sinh Ω 1x Ω 1 sinh Ω 2x , −2 ⩽ x ⩽ 2 , where Ω 1 and Ω 2 are either real or pure imaginary. If either Ω 1, or Ω 2 = 0 , one takes the appropriate limit. The associated commuting differential operator is L = d dx a(x) d dx + b(x) , with a(x) = 1 − sinh 2 Ω 2x sinh 2Ω 2 and b(x) = (Ω 1 2 − Ω 2 2) a(x) + Ω 3 . Here Ω 3 is an arbitrary real constant. Our result states that this remains true, under the relaxed assumptions that ϱ is C 2, with 0 as a non-degenerate critical point, and a( x) and b( x) are C 1 and C 0, respectively. Thus under a mild smoothness and non-degeneracy hypothesis, the only examples are those in which K is a time-and-band limiting operator and L is the operator originally found by Landau, Pollak, and Slepian.

Random walks on direct sums of discrete groups

LetG0 be a (not necessarily Abelian) discrete group, and letp0 be a probability onG0. Form the direct sumG of a countable number of copies ofG0, and letp be a probability onG which is a convex combination of copies ofp0 on the factorsG0. We consider the associated random walk onG, and study the asymptotic behavior of the probabilityp*n(e) of returning to the identity elemente aftern steps. This behavior depends heavily on the choice of the convex combination, and is considerably more complicated than the behavior for finite direct sums described in, for example, Refs. 4 and 5. A by-product of the main results is a description of the asymptotic behavior of the moments of the Cantor distribution and of the moment-generating function of symmetric Bernoulli convolutions.

Continuity of convolution semigroups on hypergroups

LetK be a commutative hypergroup with the property that either the identity character is contained in the support of the Plancherel measure onK^, or the identity character is not isolated inK^ and all characters sufficiently close (but not equal) to the identity character vanish at infinity. We present a shift compactness theorem forK and use this to prove that every symmetric convolution semigroup of probability measures onK is continuous.

A stochastic approach to quasi-everywhere boundary convergence of harmonic functions

Given a Dirichlet form E (·, ·) on the unit sphere S in R n ( n ⩾ 2) associated to a continuous, symmetric convolution semigroup of measures on a group G of isometries on S and given a ( G-invariant) Markov process X t on the open unit ball B, it is shown that for any real function u ϵ L 2( S) with E ( u, u)<∞ the X t -harmonic extension u ̃ has limit u ̌ (θ) along a.a. paths X t conditioned to exit from B at θ, for quasi-all θ ϵ S, where u ̌ is a quasi-continuous version of u. This extends in several ways classical results due to Beurling and Broman about the existence of radial limits quasi-everywhere for a harmonic function in the open unit disc in the plane with a finite Dirichlet integral.

Supports of holomorphic convolution semigroups and densities of symmetric convolution semigroups on a locally compact group

Supports of holomorphic convolution semigroups and densities of symmetric convolution semigroups on a locally compact group