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Negligible Variation, Change of Variables, and a Smooth Analog of the Hobby-Rice Theorem

This dissertation concerns two topics in analysis. The rst section is an exposition
of the Henstock-Kurzweil integral leading to a necessary and su cient condition
for the change of variables formula to hold, with implications for the change
of variables formula for the Lebesgue integral. As a corollary, a necessary and suf-
cient condition for the Fundamental Theorem of Calculus to hold for the HK integral
is obtained. The second section concerns a challenge raised in a paper by O.
Lazarev and E. H. Lieb, where they proved that, given f1….,fn ∈ L1 ([0,1] ; C),
there exists a smooth function φ that takes values on the unit circle and annihilates
span {f1...., fn}. We give an alternative proof of that fact that also shows the W1,1
norm of φ can be bounded by 5πn + 1. Answering a question raised by Lazarev and
Lieb, we show that if p > 1 then there is no bound for the W1,p norm of any such
multiplier in terms of the norms of f1...., fn. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2016. / FAU Electronic Theses and Dissertations Collection

Identiferoai:union.ndltd.org:fau.edu/oai:fau.digital.flvc.org:fau_33486
ContributorsRutherfoord, Vermont Charles (author), Sagher, Yoram (Thesis advisor), Florida Atlantic University (Degree grantor), Charles E. Schmidt College of Science, Department of Mathematical Sciences
PublisherFlorida Atlantic University
Source SetsFlorida Atlantic University
LanguageEnglish
Detected LanguageEnglish
TypeElectronic Thesis or Dissertation, Text
Format57 p., application/pdf
RightsCopyright © is held by the author, with permission granted to Florida Atlantic University to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder., http://rightsstatements.org/vocab/InC/1.0/

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