Francos, Gregory Peter (2011) *Luzin Type Approximation of Functions of Bounded Variation.* Doctoral Dissertation, University of Pittsburgh.

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## Abstract

This paper is divided into two sections:(I) Consider the function space BV^m = {u ∈ W^{m-1,1}: D^α u is a measure for |α| = m}Such functions are called mth order functions of bounded variation. We show that a function in BV^m(R^n) possesses the so-called C^m-Luzin property; that is, it coincides with a C^m(R^n) function outside a set of arbitrarily small Lebesgue measure.(II) Consider a set of Lebesgue measureable functions f^α: R^N &rarr R indexed by themulti-indices in R^N of order |α| = m. We will prove that for any such collection, there isg &isin C^{m-1}(R^N) which is m-times differentiable almost everywhere, and such thatD^α g(x) = f^α(x) a.e. for all |α| = m.

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## Details | ||||||||||||||||

Item Type: | University of Pittsburgh ETD | |||||||||||||||
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Title: | Luzin Type Approximation of Functions of Bounded Variation | |||||||||||||||

Status: | Unpublished | |||||||||||||||

Abstract: | This paper is divided into two sections:(I) Consider the function space BV^m = {u ∈ W^{m-1,1}: D^α u is a measure for |α| = m}Such functions are called mth order functions of bounded variation. We show that a function in BV^m(R^n) possesses the so-called C^m-Luzin property; that is, it coincides with a C^m(R^n) function outside a set of arbitrarily small Lebesgue measure.(II) Consider a set of Lebesgue measureable functions f^α: R^N &rarr R indexed by themulti-indices in R^N of order |α| = m. We will prove that for any such collection, there isg &isin C^{m-1}(R^N) which is m-times differentiable almost everywhere, and such thatD^α g(x) = f^α(x) a.e. for all |α| = m. | |||||||||||||||

Date: | 27 September 2011 | |||||||||||||||

Date Type: | Completion | |||||||||||||||

Defense Date: | 06 May 2011 | |||||||||||||||

Approval Date: | 27 September 2011 | |||||||||||||||

Submission Date: | 24 May 2011 | |||||||||||||||

Access Restriction: | No restriction; The work is available for access worldwide immediately. | |||||||||||||||

Patent pending: | No | |||||||||||||||

Institution: | University of Pittsburgh | |||||||||||||||

Thesis Type: | Doctoral Dissertation | |||||||||||||||

Refereed: | Yes | |||||||||||||||

Degree: | PhD - Doctor of Philosophy | |||||||||||||||

URN: | etd-05242011-140235 | |||||||||||||||

Uncontrolled Keywords: | Potential Theory; Distributions; Functions of Bounded Variation; Whitney Extension Theorem; Luzin Property; Real Analysis | |||||||||||||||

Schools and Programs: | Dietrich School of Arts and Sciences > Mathematics | |||||||||||||||

Date Deposited: | 10 Nov 2011 14:45 | |||||||||||||||

Last Modified: | 11 Jan 2012 15:19 | |||||||||||||||

Other ID: | http://etd.library.pitt.edu/ETD/available/etd-05242011-140235/, etd-05242011-140235 |

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