A'brute force'kind of numerical integration can be done, if the integrand is reasonably well-behaved ( i . e . piecewise continuous and of bounded variation ), by evaluating the integrand with very small increments.
22.
Vazif Meylanov is the author of two works on the mathematical theory of real-variable function : Sequences of closed sets of bounded variation converging in the deviation metric ( 1974 ) and Two close sets of bounded variation .
23.
The key theorem in constructing the Khinchin integral is this : a function " f " that is generalized absolutely continuous ( or even of " generalized bounded variation ", a weaker notion ) has an approximate derivative almost everywhere.
24.
However, the article on absolute continuity says it is equivalent to the Lusin N property and bounded variation . " f is absolutely continuous if and only if it is continuous, is of bounded variation and has the Luzin N property . " Right?
25.
However, the article on absolute continuity says it is equivalent to the Lusin N property and bounded variation . " f is absolutely continuous if and only if it is continuous, is of bounded variation and has the Luzin N property . " Right?
26.
In mathematics, a function of "'bounded deformation "'is a function whose distributional derivatives are not quite well-behaved-enough to qualify as functions of bounded variation, although the symmetric part of the derivative matrix does meet that condition.
27.
The hypotheses can be weakened, as in the results of Carleman and Hunt, to being, provided that is in the interior of a closed interval on which is continuous and of bounded variation, and provided that the integrals are taken in the sense of Cauchy principal values.
28.
The Riemann Stieltjes integral appears in the original formulation of F . Riesz's theorem which represents the dual space of the Banach space " C " [ " a ", " b " ] of continuous functions in an interval [ " a ", " b " ] as Riemann Stieltjes integrals against functions of bounded variation.
29.
Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation ( that assigns to each function of bounded variation the corresponding Lebesgue Stieltjes measure, and the integral with respect to the Lebesgue Stieltjes measure agrees with the Riemann Stieltjes integral for continuous functions ), the above stated theorem generalizes the original statement of F . Riesz . ( See Gray ( 1984 ), for a historical discussion ).
30.
Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation ( that assigns to each function of bounded variation the corresponding Lebesgue Stieltjes measure, and the integral with respect to the Lebesgue Stieltjes measure agrees with the Riemann Stieltjes integral for continuous functions ), the above stated theorem generalizes the original statement of F . Riesz . ( See Gray ( 1984 ), for a historical discussion ).
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