scholarly journals On the localization of discontinuities of the first kind for a function of bounded variation

2013 ◽  
Vol 280 (S1) ◽  
pp. 13-25 ◽  
Author(s):  
A. L. Ageev ◽  
T. V. Antonova
Keyword(s):  
Bounded Variation ◽  
Function Of Bounded Variation

scholarly journals On second order nonlocal boundary value problem at resonance

2016 ◽  
Vol 56 (1) ◽  
pp. 143-153 ◽  
Author(s):  
Katarzyna Szymańska-Dębowska
Keyword(s):  
Boundary Value Problem ◽  
Bounded Variation ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Second Order ◽  
Function Of Bounded Variation ◽  
Nonlocal Boundary Value Problem ◽  
At Resonance ◽  
Problem At Resonance

Abstract This work is devoted to the existence of solutions for a system of nonlocal resonant boundary value problem $$\matrix{{x'' = f(t,x),} \hfill & {x'(0) = 0,} \hfill & {x'(1) = {\int_0^1 {x(s)dg(s)},} }} $$ where f : [0, 1] × ℝk → ℝk is continuous and g : [0, 1] → ℝk is a function of bounded variation.

scholarly journals Asymptotic (statistical) periodicity in two-dimensional maps

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Fumihiko Nakamura ◽  
Michael C. Mackey
Keyword(s):  
Dynamical System ◽  
Bounded Variation ◽  
High Dimensional ◽  
Sufficient Condition ◽  
Two Dimensional ◽  
Function Of Bounded Variation ◽  
Dimensional Dynamical System ◽  
Asymptotic Periodicity ◽  
Parameter Values

<p style='text-indent:20px;'>In this paper we give a new sufficient condition for the existence of asymptotic periodicity of Frobenius–Perron operators corresponding to two–dimensional maps. Asymptotic periodicity for strictly expanding systems, that is, all eigenvalues of the system are greater than one, in a high-dimensional dynamical system was already known. Our new result enables one to deal with systems having an eigenvalue smaller than one. The key idea for the proof is to use a function of bounded variation defined by line integration. Finally, we introduce a new two-dimensional dynamical system numerically exhibiting asymptotic periodicity with different periods depending on parameter values, and discuss the application of our theorem to the example.</p>

On Density of Fourier Coefficients

1973 ◽  
Vol 16 (1) ◽  
pp. 93-103 ◽  
Author(s):  
Rafat N. Siddiqi
Keyword(s):  
Fourier Series ◽  
Total Variation ◽  
Bounded Variation ◽  
Fourier Coefficients ◽  
Function Of Bounded Variation ◽  
Fourier Sine ◽  
Image Position

Letfbe anLintegrable real valued function of period 2π and let(1)be its Fourier series. It is known that iffis of bounded variation then allnanandnbn(n=1,2,3,…) lie in the interval [-V(F)/π, V(F)/π;] whereV(f) is the total variation off. M. Izumi and S. Izumi [3] have recently asserted the following theorem A about the density of the positive and negative Fourier sine coefficients of a function of bounded variation.

scholarly journals Henstock-Kurzweil Integral Transforms

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Salvador Sánchez-Perales ◽  
Francisco J. Mendoza Torres ◽  
Juan A. Escamilla Reyna
Keyword(s):  
Bounded Variation ◽  
Integral Transforms ◽  
Function Of Bounded Variation

We show conditions for the existence, continuity, and differentiability of functions defined by , where is a function of bounded variation on with .

scholarly journals Convolutions with the Continuous Primitive Integral

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Erik Talvila
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Bounded Variation ◽  
Real Line ◽  
Fubini Theorem ◽  
Function Of Bounded Variation ◽  
Distributional Derivative ◽  
The Real ◽  
Integrable Distribution ◽  
Uniformly Continuous

IfFis a continuous function on the real line andf=F′is its distributional derivative, then the continuous primitive integral of distributionfis∫abf=F(b)−F(a). This integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals. Under the Alexiewicz norm, the space of integrable distributions is a Banach space. We define the convolutionf∗g(x)=∫−∞∞f(x−y)g(y)dyforfan integrable distribution andga function of bounded variation or anL1function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. Forgof bounded variation,f∗gis uniformly continuous and we have the estimate‖f∗g‖∞≤‖f‖‖g‖ℬ&#x1D4B1;, where‖f‖=supI|∫If|is the Alexiewicz norm. This supremum is taken over all intervalsI⊂ℝ. Wheng∈L1, the estimate is‖f∗g‖≤‖f‖‖g‖1. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.

scholarly journals On a condition that a trigonometrical series should have a certain form

1913 ◽  
Vol 88 (606) ◽  
pp. 569-574 ◽  
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Difficult Problem ◽  
Functions Of Bounded Variation ◽  
Function Of Bounded Variation ◽  
Necessary And Sufficient Condition ◽  
Trigonometrical Series ◽  
Derived Series ◽  
Necessary And Sufficient ◽  
Series Of Functions

1. In a recent communication to the Society I have illustrated the fact that the derived series of the Fourier series of functions of bounded variation play a definite part in the theory of Fourier series. Some of the more interesting theorems in that theory can only be stated in all their generality when the coefficients of such derived series take the place of the Fourier constants of a function. I have also recently shown that Lebesgue’s theorem, whether in its original or in its extended form, with regard to the usual convergence of a Fourier series when summed in the Cesàro manner is equally true for the derived series of Fourier series of functions of bounded variation. I have also pointed out that, in considering the effect of all known convergence factors in producing usual convergence, it is immaterial whether the series considered be a Fourier series, or such a derived series. We are thus led to regard the derived series of the Fourier series of functions of bounded variation as a kind of pseudo-Fourier series, possessing properly so-called. In particular we are led to ask ourselves what is the necessary and sufficient condition that a trigonometrical series should have the form in question. One answer is of course immediate. The integrated series must converge to a function of bounded variation. This is merely a statement in slightly different language of the property in question. We require a condition of a simpler formal character, one which does not require us to solve the difficult problem as to whether an assigned trigonometrical series not only converges but also has for sum a function of bounded variation.

scholarly journals ON THE POSITIVITY OF RIEMANN–STIELTJES INTEGRALS

2012 ◽  
Vol 87 (3) ◽  
pp. 400-405 ◽  
Author(s):  
JANI LUKKARINEN ◽  
MIKKO S. PAKKANEN
Keyword(s):  
Continuous Function ◽  
Bounded Variation ◽  
Nonnegative Function ◽  
Stieltjes Integral ◽  
Function Of Bounded Variation ◽  
Positive Continuous Function

AbstractWe study the question whether a Riemann–Stieltjes integral of a positive continuous function with respect to a nonnegative function of bounded variation is positive.

scholarly journals bel and Euler Summation Formulas for SBV(R)

2021 ◽  
Author(s):  
Sergio Venturini
Keyword(s):  
Continuous Function ◽  
Bounded Variation ◽  
Special Function ◽  
Real Variable ◽  
Singular Part ◽  
Summation Formula ◽  
Convergent Series ◽  
Function Of Bounded Variation ◽  
Summation Formulas ◽  
Almost Everywhere

The purpose of this paper is to show that the natural setting for various Abel and Euler-Maclaurin summation formulas is the class of special function of bounded variation. A function of one real variable is of bounded variation if its distributional derivative is a Radom measure. Such a function decomposes uniquely as sum of three components: the first one is a convergent series of piece-wise constant function, the second one is an absolutely continuous function and the last one is the so-called singular part, that is a continuous function whose derivative vanishes almost everywhere. A function of bounded variation is special if its singular part vanishes identically. We generalize such space of special function of bounded variation to include higher order derivatives and prove that the functions of such spaces admit a Euler-Maclaurin summation formula. Such a result is obtained by deriving in this setting various integration by part formulas which generalizes various classical Abel summation formulas.

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