The classical Fourier transform Fcl is an isomorphism of the Schwartz space S(Rk) onto itself. The space C∞ c (Rk) of smooth functions with compact support is dense in S(Rk), and the classical Paley-Wiener theorem characterises the image of C∞ c (R k) under Fcl as rapidly decreasing functions having an holomorphic extension to Ck of exponential type. Since Rk is self-dual, the same theorem also...

Let k be a positive integer and let Gk denote the set of all joint distributions of k-tuples (a1, . . . , ak) in a non-commutative probability space (A, φ) such that φ(a1) = · · · = φ(ak) = 1. Gk is a group under the operation of free multiplicative convolution ⊠. We identify ( Gk,⊠ ) as the group of characters of a certain Hopf algebra Y. Then, by using the log map from characters to infinites...

with respect to the Gaussian measure. Macdonald extended it from An−1 to other root systems and verified his conjecture for classical ones by means of the Selberg integrals [M1]. It was established by Opdam in [O1] in full generality using the shift operators. The integral is an important normalization constant for a k-deformation of the Hankel transform introduced by Dunkl [D]. The generalized...

It is proven that, for any soft potential φ(r) characterized by a finite Fourier transform e φ(k), the virial and energy thermodynamic routes are equivalent if the Fourier transform of the total correlation function divided by the density ρ, eh(k)/ρ, is an arbitrary function of ρβ e φ(k), where β is the inverse temperature. This class of approximations includes the mean spherical approximation ...

The aim of this study is to diagnose epileptic seizures by using different machine learning algorithms. For this purpose, the frequency components of the EEG are extracted by using the discrete wavelet transform (DWT) and parametric methods based on autoregressive (AR) model. Both these two feature extraction methods are applied to the input of machine learning classification algorithms such as...

detecting blood vessels is an important task in retinal image analysis.  the task is more challenging with the presence of bright and dark lesions in retinal images. here, a method is proposed to detect vessels in both normal and abnormal retinal fundus images based on their linear features. first, the negative impact of bright lesions is reduced by using k-means segmentation in a perceptive sp...