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Luzin Type Approximation of Functions of Bounded Variation

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
    ETD Committee:
    ETD Committee TypeCommittee MemberEmail
    Committee ChairHajlasz, Piotrhajlasz@pitt.edu
    Committee MemberBeatrous, Frankbeatrous@pitt.edu
    Committee MemberLeoni, Giovannigiovanni@andrew.cmu.edu
    Committee MemberManfredi, Juanmanfredi@pitt.edu
    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    Status: Unpublished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; Release the ETD 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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