Necessary and Sufficient Conditions in the Theory of Fourier Transforms

1931 ◽  
Vol 32 (4) ◽  
pp. 830 ◽  
Author(s):  
Andrew C. Berry
Keyword(s):  
Fourier Transforms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

scholarly journals Absolute convergence of Fourier integrals and Lipschitz classes

2021 ◽  
Vol 19 ◽  
pp. 102
Author(s):  
B.I. Peleshenko
Keyword(s):  
Absolute Convergence ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lipschitz Classes ◽  
Necessary And Sufficient ◽  
Fourier Integrals

The necessary and sufficient conditions, in terms of Fourier transforms $\hat{f}$ of functions $f \in L^1(\mathbb{R})$, are obtained for $f$ to belong to the Lipschitz classes $H^{\omega}(\mathbb{R})$ and $h^{\omega}(\mathbb{R})$.

scholarly journals On convergence of Fourier integrals and Lipschitz spaces defined with differences of fractional order

2015 ◽  
Vol 23 ◽  
pp. 75
Author(s):  
B.I. Peleshenko ◽  
T.N. Semirenko
Keyword(s):  
Fractional Order ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lipschitz Spaces ◽  
Lipschitz Classes ◽  
Necessary And Sufficient ◽  
Fourier Integrals

The necessary and sufficient conditions in terms of Fourier transforms $\hat{f}$ of functions $f\in L^1(\mathbb{R})$ are obtained for $f$ to belong to the Lipschitz classes $H^{\omega}(\mathbb{R})$, $h^{\omega}(\mathbb{R})$.

scholarly journals On the L1-convergence of Fourier transforms

1996 ◽  
Vol 60 (3) ◽  
pp. 405-420
Author(s):  
Dᾰng Vũ Giang ◽  
Ferenc Móricz
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Absolutely Continuous ◽  
Inversion Formulas ◽  
Tauberian Conditions ◽  
Tauberian Type ◽  
Necessary And Sufficient

AbstractWe study cosine and sine Fourier transforms defined by F(t):= (2/π) and (t):= (2/π), where f is L1-integrable over[0, ∞]. We also assume than F are locally absolutely continuous over [0, ∞). In particular, this is the case if both f(x) and xf(x) are (L1-integrable over [0, ∞). Motivated by the inversion formulas, we consider the partial integras Sν (f, x):= and ν(f, x):= , the modified partial integrals uν (f, x):= sν(f, x) - F(ν)(sin νx)/x and ũν(f, x):= ν(f, x) + (ν) (cos νx)/x, where ν > 0. We give necessary and sufficient conditions for(L1 [0, ∞)-convergence of uν (f) and ũν (f) as well as for the L1 [0, X]-convergence of sν (f) and ν(f) to f as ν← ∞, where 0 < X < ∞ is fixed. On the other hand, in certain cases we conclude that sν(f) and ν(f) cannot belong to (L1 [0,∞). Conequently, it makes no sense to speak of their (L1 [0, ∞)-convergence as ν ← ∞.As an intermediate tool, we use the Cesàro means of Fourier transforms. Then we prove Tauberian type results and apply Sidon type inequalities in order to obtain Tauberian conditions of Hardy-Karamata kind.We extend these results to the complex Fourier transform defined by G(t):= , where g is L1- integrable over (−∞, ∞).

scholarly journals Absolute convergence of Fourier integrals and Lipschitz classes defined with differences of fractional order

2013 ◽  
Vol 21 ◽  
pp. 145
Author(s):  
B.I. Peleshenko ◽  
T.N. Semirenko
Keyword(s):  
Fractional Order ◽  
Absolute Convergence ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lipschitz Classes ◽  
Necessary And Sufficient ◽  
Fourier Integrals

The necessary and sufficient conditions in terms of Fourier transforms $\hat{f}$ of functions $f\in L^1(\mathbb{R})$ are obtained for $f$ to belong to the Lipschitz classes $H_C^{\omega, \alpha}(\mathbb{R})$ and $h_C^{\omega, \alpha}(\mathbb{R})$, defined by differences of fractional order.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj&gt; 0 for eachj&gt; 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

Sign in / Sign up

Export Citation Format

Share Document