In the present paper a Korovkin-type theorem is proposed for the approximation operators defined by the inverse Ftransforms. These results allow us to choose between a variety of shapes to be used as atoms of the fuzzy partitions used within the F-transform’s framework. In this way we can enlarge considerably the class of F-transforms proposed recently by I. Perfilieva. The new fuzzy partitions...

We explain the orthogonal planes split (OPS) of quaternions based on the arbitrary choice of one or two linearly independent pure unit quaternions f ,g. Next we systematically generalize the quaternionic Fourier transform (QFT) applied to quaternion fields to conform with the OPS determined by f ,g, or by only one pure unit quaternion f , comment on their geometric meaning, and establish invers...

The aim of this paper is to propose a new approach to probability density function (PDF) estimation which is based on the fuzzy transform (F-transform) introduced by Perfilieva in [10]. Firstly, a smoothing filter based on the combination of the discrete direct and continuous inverse F-transform is introduced and some of the basic properties are investigated. Next, an alternative approach to PD...

Abstract. We have combined the Laplace transform and the Stieltjes transform of the form f̂(x) = R∞ 0 f(t) (xm+tm)ρ dt, m, ρ > 0 and applied it to an ordered vector space of generalized functions to which the topology of bounded convergence is assigned. Some of the order properties of the transform and its inverse are studied. Also we solve an initial value problem and compare different solution...

The Short Term Fourier Transform (STFT) is a classical linear time-frequency (T-F) representation. Despites its relative simplicity, it has become a standard tool for the analysis of non-stationary signals. Since it provides a redundant representation, it raises some issues such as (i) “optimal” window choice for analysis, (ii) existence and determination of an inverse transformation, (iii) per...

The Cayley transform of A, F = (I+A)−1 (I−A), is studied when A is a P -matrix, an M -matrix, an inverse M-matrix, a positive definite matrix, or a totally nonnegative matrix. Given a matrix A in each of these positivity classes and using the fact that the Cayley transform is an involution, properties of the ensuing factorization A = (I+F )−1 (I−F ) are examined. Specifically, it is investigate...

The Cayley transform of A, F = (I+A)−1 (I−A), is studied when A is a P -matrix, an M -matrix, an inverse M-matrix, a positive definite matrix, or a totally nonnegative matrix. Given a matrix A in each of these positivity classes and using the fact that the Cayley transform is an involution, properties of the ensuing factorization A = (I+F )−1 (I−F ) are examined. Specifically, it is investigate...

We study the non-parametric estimation of an unknown density f with support on R+ based i.i.d. sample multiplicative measurement errors. The proposed fully-data driven procedure is Mellin transform f, a regularisation inverse by spectral cut-off and data-driven model selection in order to deal upcoming bias-variance trade-off. introduce discuss further Mellin-Sobolev spaces which characterize r...

We give an explicit formula (i.e., an explicit expression for the formal stationary phase formula) for the local Fourier-Laplace transform of a formal germ of meromorphic connection of one complex variable with a possibly irregular singularity. Introduction We will denote by C((t)) (resp. C({t})) the field C[[t]][t] (resp. C{t}[t]) of formal (resp. convergent) Laurent series of the variable t, ...