In this paper we define and study the Dunkl convolution product and the Dunkl transform on spaces of distributions on R. By using the main results obtained, we study the hypoelliptic Dunkl convolution equations in the space of distributions.

The commutative neutrix convolution product of the functions xe − and xe μx + is evaluated for r, s = 0, 1, 2, . . . and all λ, μ. Further commutative neutrix convolution products are then deduced.

Up to date, the middle convolution is the most powerful tool for realizing classical groups as Galois groups over Q(t). We study the middle convolution of local systems on the punctured affine line in the setting of singular cohomology and in the setting of étale cohomology. We derive a formula to compute the topological monodromy of the middle convolution in the general case and use it to dedu...

In the present paper we present the tensor-product approximation of multidimensional convolution transform discretized via collocation-projection scheme on the uniform or composite refined grids. Examples of convolving kernels are given by the classical Newton, Slater (exponential) and Yukawa potentials, 1/‖x‖, e−λ‖x‖ and e−λ‖x‖/‖x‖ with x ∈ Rd. For piecewise constant elements on the uniform gr...

This paper continues the investigation of the use of spectral analysis on certain noncommutative nite groups|wreath product groups|in digital signal processing. We describe here the generalization of discrete cyclic convolution to convolution over these groups and show how it reduces to multiplication in the spectral domain. Finite group-based convolution is de ned in both the spatial and spect...

Commonly-preserved radial convolution is a prominent characteristic of the mammalian cerebral cortex. Endeavors from multiple disciplines have been devoted for decades to explore the causes for this enigmatic structure. However, the underlying mechanisms that lead to consistent cortical convolution patterns still remain poorly understood. In this work, inspired by prior studies, we propose and ...

BACKGROUND AND PURPOSE Prefrontal cortex, known to be a crucial region for the function of attention, is generally thought to be largely associated with the pathogenesis of attention-deficit hyperactivity disorder (ADHD). Most previous structural imaging studies of ADHD reported abnormality of grey matter volume in prefrontal region. However, volume measure is affected by the size of the interr...

We present a monotonic convolution for planar regions A and B bounded by line and circular arc segments. The Minkowski sum equals the union of the cells with positive crossing numbers in the arrangement of the convolution, as is the case for the kinetic convolution. The monotonic crossing number is bounded by the kinetic crossing number, and also by the maximum number of intersecting pairs of m...

plates and shells are the significant structural components for many engineering and industrial applications. in this study, free vibration analysis of annular plates is investigated. for this, two different numerical methods such as differential quadrature and discrete singular convolution methods have been performed for numerical simulations. frequency values have been obtained via these two ...

2. Arithmetic functions 19 2.1. Arithmetic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2. Multiplicative functions . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3. The Dirichlet convolution . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4. Examples of Dirichlet convolutions . . . . . . . . . . . . . . . . . . . 34 2.5. Möbius inversion . . . . . . . . . ....