Generalized Bounded Variation Term Paper by Nicky
Generalized Bounded Variation
A review of functions of generalized bounded variation and their Fourier series properties and behavior.
# 149423
 715 words
 5 sources
 MLA
 2011

Published
on Dec 14, 2011
in
Mathematics
(General)
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Description:
The paper outlines the general properties and considerations involved in the functions of bounded and generalized bounded variation. The paper then discusses the Fourier series and provides examples where some of the Fourier properties can be best seen.
Outline:
Introduction
Functions of Bounded and Generalized Bounded Variation  Considerations
General Properties of Functions of Generalized Bounded Variation
Fourier series
Outline:
Introduction
Functions of Bounded and Generalized Bounded Variation  Considerations
General Properties of Functions of Generalized Bounded Variation
Fourier series
From the Paper:
"As such, a function of bounded variation should be the first notion before touching on functions of generalized bounded variation and, subsequently, their connection to the Fourier series. The function of bounded variation, as defined on a set E if sup F (bi)  F (ai) < , where "the supremum is taken over all sequences {[ai, bi]} of nonoverlapping intervals with endpoints in E" . In less mathematical terms, a function is of bounded variation if its total variation is finite over a certain interval for its variables."Moving this over an interval from 0 to 2 , described as [0, 2 ] and characteristic of Fourier series, then a function f of bounded pvariation on [0, 2 ], where p<=1, and to belong to the class Vp if Vp (f) = sup { f (Ii)) p }1/p < , where sup is the supremum over all nonoverlapping subintervals of [0, 2 ].
"With L.C. Young, the concept was generalized with the introduction of a continuous function defined on the interval [0; ) and strictly increasing from 0 to . Young proposed that "the variation of a function f on [0,2] is the supremum V (f) of the sums f (Ii))."
Sample of Sources Used:
 Bruckner, Andrew. Differentiation of Real Functions (Crm Monograph, Vol 5). American Mathematical Society, Centre de Recherches Mathematiques; 2 Sub edition. May 1994
 D. Waterman, On convergence of Fourier series of functions of generalized bounded variation, Studia Math.44 (1972)
 Vyas, R.G. Properties of Functions of Generalized Bounded Variation. Mathematical Analysis. Vol. 58. 2006. Pages 9196.
 A.M. Garsia; S. Sawyer; On some classes of continuous functions with convergent Fourier series. Journal of Mathematics and Mechanics.
 Bakhalov, A.N. Waterman classes and triangular sums of double Fourier series. Analysis Mathematica, Volume 27, Number 1, January 2001
Cite this Term Paper:
APA Format
Generalized Bounded Variation (2011, December 14)
Retrieved May 10, 2021, from https://www.academon.com/termpaper/generalizedboundedvariation149423/
MLA Format
"Generalized Bounded Variation" 14 December 2011.
Web. 10 May. 2021. <https://www.academon.com/termpaper/generalizedboundedvariation149423/>