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\(\alpha\)-variation and transformation into \(C^ n\) functions. (English) Zbl 0634.26006
Connections are investigated between boundedness of the generalized variation in the sense of L. C. Young, and Lipschitz or differentiability conditions for a transformation of this function. For example, if \(s>0,\) then a continuous function f on an interval \([a,b]\) is of bounded (1/s)- variation if and only if there is a homeomorphism \(\Phi\) of \([a,b]\) onto itself such that \(f\circ \Phi \in Lip s.\) Here, if \(k<s\leq k+1,\) k a positive integer, then \(f\in Lip s\) means that there exists the derivative \(f^{(k)}\in Lip(s-k).\) The authors also give similar conditions for f to be of strongly bounded \((1/s)\)-variation.
26A45 Functions of bounded variation, generalizations
26A16 Lipschitz (Hölder) classes