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.

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‖ℬ𝒱, 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 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.

On Measures Determined by Continuous Functions that are not of Bounded Variation

1970 ◽  
Vol 13 (1) ◽  
pp. 121-124 ◽  
Author(s):  
J. H. W. Burry ◽  
H. W. Ellis
Keyword(s):  
Continuous Function ◽  
Total Variation ◽  
Bounded Variation ◽  
Real Line ◽  
Open Interval ◽  
Continuous Functions ◽  
Outer Measure ◽  
Function Of Bounded Variation ◽  
Borel Sets ◽  
Nondifferentiable Function

In [1] it was shown that a continuous function of bounded variation on the real line determined a Method II outer measure for which the Borel sets were measurable and the measure of an open interval was equal to the total variation of f over the interval. The monotone property of measures implied that if an open interval I on which f was not of bounded variation contained subintervals on which f was of finite but arbitrarily large total variation then the measure of I was infinite. Since there are continuous functions that are not of bounded variation over any interval (e.g. the Weierstrasse nondifferentiable function) the general case was not resolved.

scholarly journals On functions with derivative of bounded variation: An analogue of Banach's indicatrix theorem

1986 ◽  
Vol 29 (1) ◽  
pp. 61-68
Author(s):  
Wolfgang Stadje
Keyword(s):  
Continuous Function ◽  
Bounded Variation ◽  
Function Of Bounded Variation

A simple, but nice theorem of Banach states that the variation of a continuous function F:[a, b]→ ℝ is given by where t(y) is defined as the number of x ∈ [a, b[ for which F(x)= y (see, e.g., [1], VIII.5, Th. 3). In this paper we essentially derive a similar representation for the variation of F′ which also yields a criterion for a function to be an integral of a function of bounded variation. The proof given here is quite elementary, though long and somewhat intriciate.

scholarly journals On a Paper of Maurice Sion

1959 ◽  
Vol 11 ◽  
pp. 409-415 ◽  
Author(s):  
Mark Mahowald
Keyword(s):  
Continuous Function ◽  
Bounded Variation ◽  
Real Line ◽  
Function Of Bounded Variation ◽  
The Real ◽  
Variational Measure

Let M0 be the set of measures μ on the real line such that open sets are μ*-immeasurable. While attempting to find out whether a set μ*-measurable for all μ in Mo is mapped into a similar set by a continuous function of bounded variation, Maurice Sion develops a theory for what he calls variational measure (4). As an application of the theory, he gets conditions on a function f and a set of measures M in order that f map a set, which is μ*-measurable for all μ ∈ M, into a set of the same kind. In particular he proves for his class M2 (def. 2.5), the following theorem (4, § 8.11).

scholarly journals Generalized Baskakov Kantorovich operators

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 6131-6151
Author(s):  
P.N. Agrawal ◽  
Meenu Goyal
Keyword(s):  
Rate Of Convergence ◽  
Bounded Variation ◽  
Statistical Convergence ◽  
Simultaneous Approximation ◽  
Weighted Approximation ◽  
Function Of Bounded Variation ◽  
Convergence Properties ◽  
Almost Everywhere ◽  
Direct Results ◽  
Kantorovich Operators

In this paper, we construct generalized Baskakov Kantorovich operators. We establish some direct results and then study weighted approximation, simultaneous approximation and statistical convergence properties for these operators. Finally, we obtain the rate of convergence for functions having a derivative coinciding almost everywhere with a function of bounded variation for these operators.

scholarly journals The Basic Existence Theorem of Riemann-Stieltjes Integral

2016 ◽  
Vol 24 (4) ◽  
pp. 253-259 ◽  
Author(s):  
Kazuhisa Nakasho ◽  
Keiko Narita ◽  
Yasunari Shidama
Keyword(s):  
Continuous Function ◽  
Existence Theorem ◽  
Bounded Variation ◽  
Real Line ◽  
Closed Interval ◽  
Stieltjes Integral ◽  
Function Of Bounded Variation ◽  
Basic Properties ◽  
Finite Sequences

Summary In this article, the basic existence theorem of Riemann-Stieltjes integral is formalized. This theorem states that if f is a continuous function and ρ is a function of bounded variation in a closed interval of real line, f is Riemann-Stieltjes integrable with respect to ρ. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are based on [15], [12], [10], and [11].

A Note on Measures Determined by Continuous Functions

1972 ◽  
Vol 15 (2) ◽  
pp. 289-291 ◽  
Author(s):  
A. M. Bruckner
Keyword(s):  
Continuous Function ◽  
Total Variation ◽  
Bounded Variation ◽  
Real Variable ◽  
Continuous Functions ◽  
Residual Subset ◽  
Zero Measure ◽  
Borel Measures ◽  
Left And Right ◽  
Definition Of

Ellis and Jeffery [2] studied Borel measures determined in a certain way by real valued functions of a real variable which have finite left and right hand limits at each point. If f is such a function and is of bounded variation on an interval I, then the associated measure μf has the property that μf(I) equals the total variation of f on I. The authors then indicated in [3] how some of these measures permit the definition of generalized integrals of Denjoy type. In [1], the authors construct an example of a continuous function f, not of bounded variation, such that the associated measure μf is the zero measure. The purpose of this note is to show that "most" continuous functions give rise to the zero measure in the sense that there is a residual subset R of C[a, b] such that for each f∊R, the associated measure μf is the zero measure.

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