Distributions Of Compact Support Research Articles

The generalized Stieltjes–Poisson transform over Lebesgue spaces and distributions of compact support

The generalized Stieltjes–Poisson transform over Lebesgue spaces and distributions of compact support

Iteration of the Laplace transform over generalized functions and a Post-Widder inversion formula over distributions of compact support

In this paper, we deal with the iteration of the Laplace transform over certain spaces of generalized functions. As a consequence we prove a new Post-Widder-type inversion formula for this transform over distributions of compact support.

ScreeNOT: Exact MSE-optimal singular value thresholding in correlated noise

We derive a formula for optimal hard thresholding of the singular value decomposition in the presence of correlated additive noise; although it nominally involves unobservables, we show how to apply it even where the noise covariance structure is not a priori known or is not independently estimable. The proposed method, which we call ScreeNOT, is a mathematically solid alternative to Cattell’s ever-popular but vague scree plot heuristic from 1966. ScreeNOT has a surprising oracle property: it typically achieves exactly, in large finite samples, the lowest possible MSE for matrix recovery, on each given problem instance, that is, the specific threshold it selects gives exactly the smallest achievable MSE loss among all possible threshold choices for that noisy data set and that unknown underlying true low rank model. The method is computationally efficient and robust against perturbations of the underlying covariance structure. Our results depend on the assumption that the singular values of the noise have a limiting empirical distribution of compact support; this property, which is standard in random matrix theory, is satisfied by many models exhibiting either cross-row correlation structure or cross-column correlation structure, and also by many situations with more general, interelement correlation structure. Simulations demonstrate the effectiveness of the method even at moderate matrix sizes. The paper is supplemented by ready-to-use software packages implementing the proposed algorithm: package ScreeNOT in Python (via PyPI) and R (via CRAN).

Abelian theorems for the index 2F1-transform over distributions of compact support and generalized functions

Abelian theorems for the index 2F1-transform over distributions of compact support and generalized functions

On the generalized Stieltjes transform over weighted Lebesgue spaces and distributions of compact support

On the generalized Stieltjes transform over weighted Lebesgue spaces and distributions of compact support

An operational calculus for a Mehler-Fock type index transform on distributions of compact support

An operational calculus for a Mehler-Fock type index transform on distributions of compact support

Abelian theorems for Laplace, Mellin and Stieltjes transforms over distributions of compact support and generalized functions

Abelian theorems for Laplace, Mellin and Stieltjes transforms over distributions of compact support and generalized functions

Bézout Identity in Pseudoratoinal Transfer Functions

Coprime factorizations of transfer functions play various important roles, e.g., minimality of realizations, stabilizability of systems, etc. This paper studies the Bézout condition over the ring E′(ℝ _ ) of distributions of compact support and the ring M(ℝ _ ) of measures with compact support. These spaces are known to play crucial roles in minimality of state space representations and controllability of behaviors. We give a detailed review of the results obtained thus far, as well as discussions on a new attempt of deriving general results from that for measures. It is clarified that there is a technical gap in generalizing the result for M(ℝ _ ) to that for E′(ℝ _ ). A detailed study of a concrete example is given.

The Lambert transform over distributions of compact support, $$L^1$$-functions and Boehmian spaces

In this paper, we study the Lambert transform over distributions of compact support on $$(0,\infty )$$ . We obtain an inversion formula for this transform and we prove a Parseval-type relation for the Lambert transform of functions in $$L^1 ((0,\infty ))$$ . We also extend this transform to Boehmian spaces.

Abelian theorems for Laplace and Mehler-Fock transforms of generalized functions

The main purpose of this article is to exhibit Abelian theorems for the Laplace and the Mehler-Fock transforms of general order over distributions of compact support and over certain spaces of generalized functions.

Abelian theorems for distributional Kontorovich–Lebedev and Mehler–Fock transforms of general order

Our goal in this article is to derive Abelian theorems for the Kontorovich–Lebedev and Mehler–Fock transforms of general order over distributions of compact support and over certain spaces of generalized functions.

Nonlinear Fokker-Planck equations in super-diffusive and sub-diffusive regimes

Anomalous-diffusion phenomena are very common in nature and may be suitably described by means of nonlinear Fokker-Planck equations, characterized by specific types of nonlinear diffusion contributions. The most explored situations in the literature consist in nonlinear diffusion contributions written as a power of the probability; these are directly associated with nonextensive statistical mechanics. In this work, we investigate two special limits of such a diffusion contribution, namely, the exponential and logarithmic ones, corresponding, respectively, to superdiffusive and subdiffusive regimes. An H-theorem is proven, relating these Fokker-Planck equations to entropic forms; moreover, the stationary states of these equations are shown to coincide with the equilibrium states, obtained by extremization of the associated entropic forms. Equilibrium distributions are computed in particular cases, being mostly characterized by long tails in the exponential case, whereas compact-support distributions always appear in the logarithmic one. The present results enlarge the applicability of nonlinear Fokker-Planck equations to a wider range of anomalous-diffusion phenomena, particularly those in special limits of super- and subdiffusion regimes.

Interpolation of holomorphic functions and surjectivity of Taylor coefficient multipliers

We prove criteria for global analytic solvability for some Euler type partial differential equations — a class of linear partial differential equations with variable (polynomial) coefficients. This is based on a very general Mellin type theory developed in the present paper which is valid for analytic functionals (in particular, distributions of compact support) substituting the classical Mellin transform. As a consequence, we get a characterization of moment sequences of analytic functionals by interpolation of holomorphic functions satisfying some restrictive growth conditions. This allows to study the so-called Hadamard type multipliers on real analytic functions of several variables, i.e., operators for which monomials are eigenvectors. We characterize multiplier sequences (i.e., sequences of eigenvalues of multipliers) by interpolation and describe the image of some multipliers, especially, the image of the principal examples of Hadamard multipliers: Euler type partial differential operators. We also get some results showing that the image remains unchanged after some perturbation of the original Hadamard type operator.

MaxEnt, second variation, and generalized statistics

There are two kinds of Tsallis-probability distributions: heavy tail ones and compact support distributions. We show here, by appeal to functional analysis’ tools, that for lower bound Hamiltonians, the second variation’s analysis of the entropic functional guarantees that the heavy tail q-distribution constitutes a maximum of Tsallis’ entropy. On the other hand, in the compact support instance, a case by case analysis is necessary in order to tackle the issue.

Note on Boehmians for Class of Optical Fresnel Wavelet Transforms

We extend the Fresnel-wavelet transform to the context of generalized functions, namely, Boehmians. At first, we study the Fresnel-wavelet transform in the sense of distributions of compact support. Based on this concept, we introduce two new spaces of Boehmians and proving certain related results. Further, we show that the extended transform establishes a linear and an isomorphic mapping between the Boehmian spaces. Moreover, conditions of continuity of the extended transform and its inverse with respect toδand Δ convergence are discussed in some details.

The Wave Front Set of the Wigner Distribution and Instantaneous Frequency

We prove a formula expressing the gradient of the phase function of a function f:ℝd↦ℂ as a normalized first frequency moment of the Wigner distribution for fixed time. The formula holds when f is the Fourier transform of a distribution of compact support, or when f belongs to a Sobolev space Hd/2+1+e(ℝd) where e>0. The restriction of the Wigner distribution to fixed time is well defined provided a certain condition on its wave front set is satisfied. Therefore we first need to study the wave front set of the Wigner distribution of a tempered distribution.

A variant of the Stieltjes transform on distributions of compact support

In this paper, by means of a generalized Lambert transform and the Möbius numbers, we obtain an inversion formula for a variant of the Stieltjes transform introduced by Goldberg [R.R. Goldberg, An inversion of the Stieltjes transform, Pacific J. Math. 8 (1958) 213–217.] and defined by G ( x ) = ∫ 0 ∞ f ( t ) t x 2 + t 2 d t , x > 0 on the space E ′ ( ( 0 , ∞ ) ) of distributions of compact support. We also study the analyticity of G.

The generalized Lambert transform on distributions of compact support

In this paper we study the generalized Lambert transform introduced by Goldberg (Duke Math. J. 25 (1958) 459–476) and defined by F(x)= ∫ 0 ∞ f(t) ∑ k=1 ∞ a ke −kxt dt, x>0, on the space E′((0,∞)) of distributions of compact support. An inversion formula is obtained over this space.

Applied hydrodynamic wave-resistance computation by Fourier transform

An applied Fourier transform computation for the hydrodynamic wave-resistance coefficient is shown, oriented to potential flows with a free surface and infinity depth. The presence of a ship-like body is simulated by its equivalent pressure disturbance imposed on the un-perturbed free surface, where a linearized free surface condition is used. The wave-resistance coefficient is obtained from the wave-height downstream. Two examples with closed solutions are considered: a submerged dipole, as a test-case, and a parabolic pressure distribution of compact support. In the three dimensional case, a dispersion relation is included which is a key resource for an inexpensive computation of the wave pattern far downstream like fifteen ship-lengths.

A Convolution Algebra of Delay-Differential Operators and a Related Problem of Finite Spectrum Assignability

In this paper we investigate a ring of delay-differential operators, which appears in the study of time-delay systems with commensurate point delays. Besides point-delay differential operators, this ring also contains certain distributed delays. It can be described as a certain algebraic extension of the ring of point-delay differential operators as well as a convolution algebra of compact support distributions. It will be shown that the finite spectrum assignability problem can be solved for spectrally controllable systems with controllers from this algebra.