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 | US
Published on Dec 14, 2011 in Mathematics (General)

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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.

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 p-variation 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 91-96.
  • 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

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