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Preliminaries
Published in Aliakbar Montazer Haghighi, Indika Wickramasinghe, Probability, Statistics, and Stochastic Processes for Engineers and Scientists, 2020
Aliakbar Montazer Haghighi, Indika Wickramasinghe
As a generalization of the Lebesgue and Riemann integrals, we consider the Lebesgue–Stieltjes measure,which is a regular Borel measure. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to the Lebesgue–Stieltjes measure. To define the Lebesgue–Stieltjes integral, let μ:[a,b]→ℝ be a Borel measure and bounded. Also, let f:[a,b]→ℝ be a bounded variation in [a, b]. By a function of bounded variation, it is meant a real-valued function whose total variation is bounded (finite), for instance, functions of bounded variation of a single variable differentiable at almost every point of their domain of definition. Then, the Lebesgue–Stieltjes integral is defined as ∫abμ(x)df(x).
Applied Analysis
Published in Nirdosh Bhatnagar, Introduction to Wavelet Transforms, 2020
A function f (·) is of bounded variation in every finite open interval if and only if f(x) is bounded and possesses a finite number of relative maximum and minimum values and discontinuities. That is, the function can be represented as a curve of finite length in any finite interval.
Piecewise uniform regularization for the inverse problem of microtomography with a-posteriori error estimate
Published in Inverse Problems in Science and Engineering, 2020
Alexander S. Leonov, Yanfei Wang, Anatoly G. Yagola
We shall assume for simplicity that . Suppose that the function is continuous everywhere in Π except on a set , of unknown discontinuity points. We assume that the set G has a zero 2D measure. Given the data of our inverse problem, we wish to construct a function , such that as uniformly on any closed subset , containing no discontinuity points of the function . The procedure of constructing such functions is called piecewise uniform regularization. The piecewise uniform regularization in N-dimensional case has been studied theoretically in [14–18]. In these investigations, the functions of bounded variation were widely used. Note that in the N-dimensional case the concept of the variation of a function can be represented in several ways. The properties of corresponding functions of bounded variation depend essentially on the particular construction of variation and, in general, can differ from some standard properties in the one-dimensional case. Therefore, not all constructions of N-dimensional variation are theoretically suitable for piecewise-uniform regularization. For instance, the application of the BV variation [13] to the Tikhonov regularization only ensures convergence in of the approximate solutions (see, e.g. [12]). Also, a counterexample is presented in [16] demonstrating that, in general, the use of BV variation does not guarantee pointwise (and therefore, piecewise-uniform) convergence. For piecewise uniform regularization, the most suitable is the construction of the VH variation. Readers can learn more about the theory of VH variation in [14–17]. Here, for brevity, we give only the main facts without proofs.