Schwartz Distributions Research Articles

On a Stability of Logarithmic‐Type Functional Equation in Schwartz Distributions

We prove the Hyers‐Ulam stability of the logarithmic functional equation of Heuvers and Kannappan f(x + y) − g(xy) − h(1/x + 1/y) = 0, x, y > 0, in both classical and distributional senses. As a classical sense, the Hyers‐Ulam stability of the inequality |f(x + y) − g(xy) − h(1/x + 1/y)| ≤ ϵ, x, y > 0 will be proved, where f, g, h : ℝ+ → ℂ. As a distributional analogue of the above inequality, the stability of inequality ∥u∘(x + y) − v∘(xy) − w∘(1/x + 1/y)∥ ≤ ϵ will be proved, where u, v, w ∈ 𝒟′(ℝ+) and ∘ denotes the pullback of distributions.

The prime number theorem for Beurling's generalized numbers. New cases

This paper provides new cases of the prime number theorem for Beurling’s generalized prime numbers. Let N be the distribution of a generalized number system and let π be the distribution of its primes. It is shown that N(x) = ax + O(x/ log x) (C), γ > 3/2, where (C) stands for the Cesaro sense, is sufficient for the prime number theorem to hold, i.e., π(x) ∼ x/ log x. The Cesaro asymptotic estimate explicitly means that ∫ x 1 N(t)− at t ( 1− t x )m dt = O ( x log x ) , for somem ∈ N. Therefore, it includes Beurling’s classical condition. We also show that under these conditions the Mobius function, associated to the generalized number system, has mean value equal to 0. The methods of this article are based on complex Tauberian theorems for local pseudo-function boundary behavior and arguments from the theory of asymptotic behavior of Schwartz distributions.

On Tauber's second Tauberian theorem

We study Tauberian conditions for the existence of Cesaro limits in terms of the Laplace transform. We also analyze Tauberian theorems for the existence of distributional point values in terms of analytic representations. The development of these theorems is parallel to Tauber's second theorem on the converse of Abel's theorem. For Schwartz distributions, we obtain extensions of many classical Tauberians for Cesaro and Abel summability of functions and measures. We give general Tauberian conditions in order to guarantee $(\mathrm{C},\beta)$ summability for a given order $\beta$. The results are directly applicable to series and Stieltjes integrals, and we therefore recover the classical cases and provide new Tauberians for the converse of Abel's theorem where the conclusion is Cesaro summability rather than convergence. We also apply our results to give new quick proofs of some theorems of Hardy-Littlewood and Szasz for Dirichlet series.

Composition with distributions of Wiener‐Poisson variables and its asymptotic expansion

AbstractIn this paper, we introduce the compositions of smooth Wiener‐Poisson functional and Schwartz distribution by using Malliavin calculus jump type. We also prove Watanabe's theorem in the jump‐diffusion case, that is, the asymptotic expansion formula for the compositions of a smooth Wiener‐Poisson functional with Schwartz distributions.

EXTENDED RIDGELET TRANSFORM ON DISTRIBUTIONS AND BOEHMIANS

The ridgelet transform is extended to the space of Schwartz distributions and to the space of C∞-Boehmians consistent with the classical Ridgelet transform on the space of square integrable Boehmians. The properties of the ridgelet transform like linearity, injectivity, surjectivity and continuity with respect to two notions of convergence are established. It is also justified that this extended ridgelet transform on C∞-Boehmians properly generalizes the earlier extensions of the ridgelet transform.

An axiomatic approach to the nonlinear theory of generalized functions and consistency of Laplace transforms

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application, we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the nonlinear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theory of distributions.

Distributional properties of exponential functionals of Lévy processes

We study the distribution of the exponential functional $I(\xi,\eta)=\int_0^{\infty} \exp(\xi_{t-}) \d \eta_t$, where $\xi$ and $\eta$ are independent L\'evy processes. In the general setting using the theories of Markov processes and Schwartz distributions we prove that the law of this exponential functional satisfies an integral equation, which generalizes Proposition 2.1 in Carmona et al "On the distribution and asymptotic results for exponential functionals of Levy processes". In the special case when $\eta$ is a Brownian motion with drift we show that this integral equation leads to an important functional equation for the Mellin transform of $I(\xi,\eta)$, which proves to be a very useful tool for studying the distributional properties of this random variable. For general L\'evy process $\xi$ ($\eta$ being Brownian motion with drift) we prove that the exponential functional has a smooth density on $\r \setminus \{0\}$, but surprisingly the second derivative at zero may fail to exist. Under the additional assumption that $\xi$ has some positive exponential moments we establish an asymptotic behaviour of $\p(I(\xi,\eta)>x)$ as $x\to +\infty$, and under similar assumptions on the negative exponential moments of $\xi$ we obtain a precise asympotic expansion of the density of $I(\xi,\eta)$ as $x\to 0$. Under further assumptions on the L\'evy process $\xi$ one is able to prove much stronger results about the density of the exponential functional and we illustrate some of the ideas and techniques for the case when $\xi$ has hyper-exponential jumps.

An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application, we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the non-linear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theory of distributions.

ON THE HYERS-ULAM-RASSIAS STABILITY OF THE JENSEN EQUATION IN DISTRIBUTIONS

We consider the Hyers-Ulam-Rassias stability problem 2u ◦ A 2 u ◦ P1 u ◦ P2 � (j xj p + j yj p ); x;y 2 R n for the Schwartz distributions u, which is a distributional version of the Hyers-Ulam-Rassias stability problem of the Jensen functional equation � � 2f ( x + y 2 ) f(x) f(y) � � � (j xj p + j yj p ); x;y 2 R n for the function f : R n ! C.

Поліноміальні повільно зростаючі розподіли

In the article, polynomial (nonlinear) analogue of tempered Schwartz distributions is constructed. Generalized operation of differentiation in the space of polynomial generalized functions as well as Fourier-Laplace transformation of such distributions are considered. Some examples are given.

Integral representation of Dirac distribution using the Tsallis q-exponential function

Jauregui and Tsallis have recently given a new integral representation of the Dirac distribution in terms of q-exponential function, generalizing the well-known representation in plane waves. We prove rigorously this result using the analytical representation of Schwartz distributions and the tools of complex analysis.

On the Point Behavior of Fourier Series and Conjugate Series

We investigate the point behavior of periodic functions and Schwartz distributions when the Fourier series and the conjugate series are both Abel summable at a point. In particular we show that if f is a bounded function and its Fourier series and conjugate series are Abel summable to values γ and β at the point θ_0 , respectively, then the primitive of f is differentiable at θ_0 , with derivative equal to γ , the conjugate function satisfies \lim_{θ→θ_0} \frac{3}{(θ–θ_0)^3} \int^θ_{θ_0} \tilde f(t)(θ–t)^2 dt = β , and the Fourier series and the conjugate series are both (\mathrm{C}, κ) summable at θ_0 , for any κ > 0 . We show a similar result for positive measures and L^1 functions bounded from below. Since the converse of our results are valid, we therefore provide a complete characterization of simultaneous Abel summability of the Fourier and conjugate series in terms of “average point values”, within the classes of positive measures and functions bounded from below. For general L^1 functions, we also give a.e. distributional interpretation of − \frac{1}{2π} p.v. \int^π_{–π} f (t+θ_0) \cot \frac t2 dt as the point value of the conjugate series when viewed as a distribution. We obtain more general results of this kind for arbitrary trigonometric series with coefficients of slow growth, i.e., periodic distributions.

Topological vector spaces problems in homology and homotopy theories

Three data are interesting here: domains of integration, integrands and integration itself. There is a lack of symmetry between polyhedral chains as domains of integration and differential forms as integrands. The non-symmetric situation disappears after considering the topological spaces of the de Rham differential forms and forms with compact supports and their strong duals, i.e., currents with compact supports and currents, respectively. This idea goes back to Schwartz distributions and Schwartz distributions with compact supports, in other terminology, generalized functions and generalized functions with compact supports. Some problems are raised, e.g., whether every quasi-complete barreled nuclear space E, whose strongly dual E ′ is nuclear, is strongly hereditary reflexive. This concerns the above mentioned de Rham spaces. Problems on R - and Q -homotopy, proper R - and Q -homotopy and proper R - and Q -homotopy at infinity are also considered as well as the coalgebra structure on currents and currents with compact supports. The classical theorem concerning derivation of additive functions with respect to volumes in points is generalized to a theorem on derivation of continuous m-forms with compact supports ω m of an oriented n-dimensional C 1 -manifold M n with respect to its m-dimensional oriented submanifolds V m in compact regular oriented submanifolds L k of M n , 0 ⩽ k < m ⩽ n .

Discrete Schwartz distributions and the Riemann zeta-function

We obtain criteria for the Riemann hypothesis using discrete Schwartz distributions defined by the arithmetical functions of Euler, von Mangoldt, Mobius, and Liouville.

Wavelet expansions and asymptotic behavior of distributions

We develop a distribution wavelet expansion theory for the space of highly time-frequency localized test functions over the real line S 0 ( R ) ⊂ S ( R ) and its dual space S 0 ′ ( R ) , namely, the quotient of the space of tempered distributions modulo polynomials. We prove that the wavelet expansions of tempered distributions converge in S 0 ′ ( R ) . A characterization of boundedness and convergence in S 0 ′ ( R ) is obtained in terms of wavelet coefficients. Our results are then applied to study local and non-local asymptotic properties of Schwartz distributions via wavelet expansions. We provide Abelian and Tauberian type results relating the asymptotic behavior of tempered distributions with the asymptotics of wavelet coefficients.

Eigenvalue Conditions for Convergence of Singularly Perturbed Matrix Exponential Functions

We investigate convergence of sequences of $n\times n$ matrix exponential functions $t\rightarrow e^{tA_{k}^{-1}}$ for $t>0$, where $A_{k}\rightarrow A$, $A_{k}$ is nonsingular and A is nilpotent. Specifically, we address pointwise convergence, almost uniform convergence, and, viewing the exponential as a Schwartz distribution, weak$^{\ast}$ convergence. We show that simple results can be obtained in terms of the eigenvalues of $A_{k}^{-1}$ alone. In particular, a necessary and sufficient condition for weak$^{\ast}$ convergence in terms of eigenvalue behavior is attainable. We then apply our results to real-analytic matrices $A(\varepsilon)$ as $\varepsilon\rightarrow0^{+}$. Our work is applicable to matrices over both $\mathbb{R}$ and $\mathbb{C}$.

Stability of Trigonometric Functional Equations in Generalized Functions

We consider the Hyers-Ulam stability of a class of trigonometric functional equations in the spaces of generalized functions such as Schwartz distributions, Fourier hyperfunctions, and Gelfand generalized functions.

Stability of a generalized quadratic functional equation in Schwartz distributions

We consider the Hyers-Ulam stability problem of the generalized quadratic functional equation % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipC0xg9qqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaGaemyDauNaeSigI8MaemyqaeueduuDJXwAKbYu51My % VXgaiyaacqWFRaWkcqWG2bGDcqWIyiYBcqWGcbGqcqWFsislcqWFYa % GmcqWG3bWDcqWIyiYBcqWGqbaudaWgaaWcbaGae8xmaedabeaakiab % -jHiTiab-jdaYmaaBaaaleaaiyGacqGF6oWAaeqaaOGaeSigI8Maem % iuaa1aaSbaaSqaaiab-jdaYaqabaGccqWF9aqpcqWFWaamcqWFSaal % aaa!5886! $$ u \circ A + v \circ B - 2w \circ P_1 - 2_\kappa \circ P_2 = 0, $$ which is a distributional version of the classical generalized quadratic functional equation % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipC0xg9qqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaGaemOzayweduuDJXwAKbYu51MyVXgaiyaacqWFOaak % cqWG4baEcqWFRaWkcqWG5bqEcqWFPaqkcqWFRaWkcqWGNbWzcqWFOa % akcqWG4baEcqWFsislcqWG5bqEcqWFPaqkcqWFsislcqWFYaGmcqWG % ObaAcqWFOaakcqWG4baEcqWFPaqkcqWFsislcqWFYaGmcqWGRbWAcq % WFOaakcqWG5bqEcqWFPaqkcqWF9aqpcqWFWaamcqWFUaGlaaa!5D5C! $$ f(x + y) + g(x - y) - 2h(x) - 2k(y) = 0. $$

On the Extension of Schwartz Distributions to the Space of Discontinuous Test Functions of Several Variables

On the Extension of Schwartz Distributions to the Space of Discontinuous Test Functions of Several Variables

ON A STABILITY OF PEXIDERIZED EXPONENTIAL EQUATION

We prove the Hyers-Ulam stability of a Pexiderized exponential equation of mappings f, g, h : <TEX>$G{\times}S{\rightarrow}{\mathbb{C}}$</TEX>, where G is an abelian group and S is a commutative semigroup which is divisible by 2. As an application we obtain a stability theorem for Pexiderized exponential equation in Schwartz distributions.