| Some applications of the Discrete Fourier Transform (DFT) in coding and in cryptography are described. The DFT over general commutative rings is introduced and the condition for its existence given. Blahut's Theorem, which relates the DFT to linear complexity, is shown to hold unchanged in general commutative rings. I. The (Usual) Discrete Fourier Transform Let be a primitive N th root of uni...

Formulæ are derived for expressing Cauchy and Hilbert transforms of a function f in terms of Cauchy and Hilbert transforms of f(xr). When r is an integer, this corresponds to evaluating the Cauchy transform of f(xr) at all choices of z1/r. Related formuæ for rational r result in a reduction to a generalized Cauchy transform living on a Riemann surface, which in turn is reducible to the standard...

In this paper, we will illustrate the F-transform based on generalized fuzzy partitions as a tool for expectile smoothing. This allows to represent a time series in terms of a fuzzy-valued function whose level-cuts are modeled by F-transform and estimated by expectile regression. The proposed methodology is illustrated on real economic and …nancial time series. Keywords: Fuzzy Transform, Expect...

the necessary conditions to transform hydroxyamide (1) and 2-oxazolines (2, 3) into 3- or 4-methyl benzo[f] isoquinolines (4, 5) has been established. the mechanism of these reactions has been proposed claiming the existence of the protonated 2-oxazoline (6) and the styrylamides (7) as intermediates.

The purpose of this paper is to present an algorithm for evaluating Han-kel transform of the null and the first kind. The result is the exact analytical representation as the series of the Bessel and Struve functions multiplied by the wavelet coefficients of the input function. Numerical evaluation of the test function with known analytical Hankel transform illustrates the proposed algorithm. T...

We show a pointwise estimate for the Fourier transform on the line involving the number of times the function changes monotonicity. The contrapositive of the theorem may be used to find a lower bound to the number of local maxima of a function. We also show two applications of the theorem. The first is the two weight problem for the Fourier transform, and the second is estimating the number of ...

Abstract. The aim of this paper is to establish an analogue of Logvinenko-Sereda’s theorem for the Fourier-Bessel transform (or Hankel transform) Fα of order α > −1/2. Roughly speaking, if we denote by PWα(b) the Paley-Wiener space of L -functions with FourierBessel transform supported in [0, b], then we show that the restriction map f → f |Ω is essentially invertible on PWα(b) if and only if Ω...

The paper is devoted to the smoothing of discrete functions using the fuzzy transform (shortly, Ftransform) introduced by Perfilieva in [1]. We generalize a smoothing filter based on the fuzzy transform proposed in [2] to obtain a better control on the smoothed functions. For this purpose, a generalization of the concept of fuzzy partition is suggested and the smoothing filter is defined as a c...