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Functions of bounded variation on compact subsets of the plane

Brenden Ashton, Ian Doust (2005)

Studia Mathematica

A major obstacle in extending the theory of well-bounded operators to cover operators whose spectrum is not necessarily real has been the lack of a suitable variation norm applicable to functions defined on an arbitrary nonempty compact subset σ of the plane. In this paper we define a new Banach algebra BV(σ) of functions of bounded variation on such a set and show that the function-theoretic properties of this algebra make it better suited to applications in spectral theory than those used previously....

Functions of Bounded Variation on Idempotent Semigroups:

Richard A. ALÒ, Andre de KORVIN (1971)

Mathematische Annalen

Functions of bounded variation, signed measures, and a general Koksma-Hlawka inequality

Christoph Aistleitner, Josef Dick (2015)

Acta Arithmetica

We prove a correspondence principle between multivariate functions of bounded variation in the sense of Hardy and Krause and signed measures of finite total variation, which allows us to obtain a simple proof of a generalized Koksma-Hlawka inequality for non-uniform measures. Applications of this inequality to importance sampling in Quasi-Monte Carlo integration and tractability theory are given. We also discuss the problem of transforming a low-discrepancy sequence with respect to the uniform measure...

Functions of finite fractional variation and their applications to fractional impulsive equations

Dariusz Idczak (2017)

Czechoslovak Mathematical Journal

We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak $σ$ -additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a $σ$ -additive term—we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e. equations containing the Dirac measures.

Functions of several variables of finite variation and their differentiability

Dariusz Idczak (1994)

Annales Polonici Mathematici

Some differentiability properties of functions of several variables of finite variation are investigated.