scholarly journals Two-Point Quadrature Rules for Riemann–Stieltjes Integrals with Lp–error estimates

2018 ◽  
Vol 4 (2) ◽  
pp. 94-109
Author(s):  
M.W. Alomari
Keyword(s):  
Error Estimates ◽  
Bounded Variation ◽  
Quadrature Rules ◽  
Stieltjes Integral ◽  
Hölder Condition ◽  
Sharp Error

AbstractIn this work, we construct a new general two-point quadrature rules for the Riemann–Stieltjes integral $\int_a^b {f(t)} \,du\,(t)$, where the integrand f is assumed to be satisfied with the Hölder condition on [a, b] and the integrator u is of bounded variation on [a, b]. The dual formulas under the same assumption are proved. Some sharp error Lp–Error estimates for the proposed quadrature rules are also obtained.

THE EFFECT OF THE WEYL FRACTIONAL INTEGRAL ON FUNCTIONS

Fractals ◽  
2021 ◽  
Author(s):  
XIA TING ◽  
CHEN LEI ◽  
LUO LING ◽  
WANG YONG
Keyword(s):  
Linear Relationship ◽  
Bounded Variation ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Fractional Integrals ◽  
Hölder Condition ◽  
Weierstrass Function ◽  
Variation Function ◽  
Weyl Fractional Integral

This paper mainly discusses the influence of the Weyl fractional integrals on continuous functions and proves that the Weyl fractional integrals can retain good properties of many functions. For example, a bounded variation function is still a bounded variation function after the Weyl fractional integral. Continuous functions that satisfy the Holder condition after the Weyl fractional integral still satisfy the Holder condition, furthermore, there is a linear relationship between the order of the Holder conditions of the two functions. At the end of this paper, the classical Weierstrass function is used as an example to prove the above conclusion.

On Ward's Perron-Stieltjes Integral

1957 ◽  
Vol 9 ◽  
pp. 96-109 ◽  
Author(s):  
Ralph Henstock
Keyword(s):  
Bounded Variation ◽  
Large Set ◽  
Stieltjes Integral ◽  
Countable Number ◽  
Finite Function ◽  
Set Of Points

In the paper (5), Ward defines an integral of Perron type of a finite function f with respect to another finite function g, where g need not be of bounded variation. There arise two problems, (a) and (b) below, that have not been dealt with in (5).If f = j at a countable number of points everywhere dense in (a, b), where f and j are both integrable with respect to g, then f — j can be nonzero on a large set of points of (a, b).

scholarly journals Error estimate for optimality of distributed parameter control problems via duality

1995 ◽  
Vol 8 (2) ◽  
pp. 177-188
Author(s):  
W. L. Chan ◽  
S. P. Yung
Keyword(s):  
Error Estimate ◽  
Error Estimates ◽  
Boundary Control ◽  
Impulsive Control ◽  
Hyperbolic Systems ◽  
Distributed Parameter ◽  
Control Problems ◽  
Parameter Control ◽  
Distributed Parameter Control ◽  
Sharp Error

Sharp error estimates for optimality are established for a class of distributed parameter control problems that include elliptic, parabolic, hyperbolic systems with impulsive control and boundary control. The estimates are obtained by constructing manageable dual problems via the extremum principle.

scholarly journals Two-point formulae of Euler type

2002 ◽  
Vol 44 (2) ◽  
pp. 221-245 ◽  
Author(s):  
M. Matić ◽  
C. E. M. Pearce ◽  
J. Pečarić
Keyword(s):  
Error Estimates ◽  
Bounded Variation

AbstractAn analysis is made of quadrature viatwo-point formulae when the integrand is Lipschitz or of bounded variation. The error estimates are shown to be as good as those found in recent studies using Simpson (three-point) formulae.

The existence of parametric surface integrals

1960 ◽  
Vol 1 (4) ◽  
pp. 419-427
Author(s):  
J. H. Michael
Keyword(s):  
Bounded Variation ◽  
Continuous Mapping ◽  
Measurable Subset ◽  
Signed Measure ◽  
Stieltjes Integral ◽  
Parametric Surface ◽  
Surface Integral ◽  
Linear Type ◽  
Definition Of ◽  
Image Position

In [2] we studied parametric n-surfaces (f, Mn), where Mn was a compact, oriented, topological n-manifold and f a continuous mapping of Mn into the real euclidean k-space Rn (k≧n). A definition of bounded variation was given and, for each surface with bounded variation and each projection P from Rk to Rn, a signed measure: Was constructed. This measure was used to define a linear type of surface integral: over a “measurable” subset A of Mn, as the Lebesgue-Stieltjes integral: .

Some sharp error estimates

2018 ◽  
pp. 473-494
Author(s):  
Doina Cioranescu ◽  
Alain Damlamian ◽  
Georges Griso
Keyword(s):  
Error Estimates ◽  
Sharp Error
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