Some generalizations of the notion of bounded variation.

*(English)*Zbl 1155.26008The definition of real valued function of a real variable that are of bounded variation was introduced by Jordan in 1881 and gave a linear class of functions whose Fourier series converge everywhere to the correct value, the Dirichlet-Jordan theorem. This excellent paper traces various extensions of this concept and just how these extensions relate to the Dirichlet-Jordan Theorem.

The first extension, by Wiener in 1924, replaces the sums in the orginal definition, \(\sum| f(I)| \), by \(\sum| f(I)| ^2\) and the Dirichlet-Jordan Theorem was shown to hold for this class of functions of bounded quadratic variation. L. C. Young and Love extended this to bounded \(p\)-variation functions, \(p\geq 1\), 1937. The extension to the class where the sum is \(\sum\Phi(| f(I)|)\), \(\Phi\) a continuous strictly increasing function, attempted by Young at the same time was more troublesome and it was not until 1940 that Salem identified the condition on the function \(\Phi\) that would ensure the validity of the Dirichlet-Jordan theorem for functions of bounded \(\Phi\)-variation. In 1972 Waterman defined a completely different extension by considering \(\sum_n| f(I_n)| / \lambda_n\) where \(\Lambda=\{\lambda_1,\lambda_2, \dots \}\) is an increasing sequence of positive numbers with \(\sum1/ \lambda_n = \infty\). Waterman showed that if \(\Lambda=\{1,2, \dots \}\) then the Dirichlet-Jordan Theorem holds and further it does not hold for any larger class of the type he defined.

All this and the relationships between these various extensions is discussed in detail in this paper.

The first extension, by Wiener in 1924, replaces the sums in the orginal definition, \(\sum| f(I)| \), by \(\sum| f(I)| ^2\) and the Dirichlet-Jordan Theorem was shown to hold for this class of functions of bounded quadratic variation. L. C. Young and Love extended this to bounded \(p\)-variation functions, \(p\geq 1\), 1937. The extension to the class where the sum is \(\sum\Phi(| f(I)|)\), \(\Phi\) a continuous strictly increasing function, attempted by Young at the same time was more troublesome and it was not until 1940 that Salem identified the condition on the function \(\Phi\) that would ensure the validity of the Dirichlet-Jordan theorem for functions of bounded \(\Phi\)-variation. In 1972 Waterman defined a completely different extension by considering \(\sum_n| f(I_n)| / \lambda_n\) where \(\Lambda=\{\lambda_1,\lambda_2, \dots \}\) is an increasing sequence of positive numbers with \(\sum1/ \lambda_n = \infty\). Waterman showed that if \(\Lambda=\{1,2, \dots \}\) then the Dirichlet-Jordan Theorem holds and further it does not hold for any larger class of the type he defined.

All this and the relationships between these various extensions is discussed in detail in this paper.

Reviewer: Peter S. Bullen (Vancouver)

##### MSC:

26A45 | Functions of bounded variation, generalizations |