# Read PDF Differentiability of Six Operators on Nonsmooth Functions and p-Variation

## Estimating the p-Variation Index of a Sample Function: An Application to Financial Data Set

The book is about differentiability of six operators on functions or pairs of functions: composition f of g , integration of f dg , multiplication and convolution of two functions, both varying, and the product integral and inverse operators for one function. The operators are differentiable with respect to p-variation norms with optimal remainder bounds. Thus the functions as arguments of the operators can be nonsmooth, possibly discontinuous, but four of the six operators turn out to be analytic holomorphic for some p-variation norms.

The reader will need to know basic real analysis, including Riemann and Lebesgue integration.

- Richard M. Dudley.
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The book is intended for analysts, statisticians and probabilists. Analysts and statisticians have each studied the differentiability of some of the operators from different viewpoints, and this volume seeks to unify and expand their results. Dudley , R. The book is about differentiability of six operators on functions or pairs of functions: composition f of g , integration of f dg , multiplication and convolution of two functions, both varying, and the product integral and inverse operators for one function.

go The operators are differentiable with respect to p-variation norms with optimal remainder bounds. Thus the functions as arguments of the operators can be nonsmooth, possibly discontinuous, but four of the six operators turn out to be analytic holomorphic for some p-variation norms.

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The reader will need to know basic real analysis, including Riemann and Lebesgue integration. The book is intended for analysts, statisticians and probabilists.

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Analysts and statisticians have each studied the differentiability of some of the operators from different viewpoints, and this volume seeks to unify and expand their results. The twofunction composition operator FG Fo G.

The integration operator and Young integrals. The product integral.