Interpolation and Approximation of Non Differentiable Function Using Polynomial

In any approximation problem, we concerned with a measure of closeness of polynomial ()xp to a function ()xf. Approximations are often obtained by power series expansions in which the higher order terms are dropped. One of the fundamental idea in the differential calculus is that a function can be...

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Main Author:	Koo, Lee Feng
Format:	Thesis
Language:	English
Published:	2007
Subjects:	Interpolation Approximation theory Nondifferentiable functions
Online Access:	http://psasir.upm.edu.my/id/eprint/5002/1/FS_2007_19.pdf
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id	my-upm-ir.5002
record_format	uketd_dc
spelling	my-upm-ir.50022015-05-20T08:15:21Z Interpolation and Approximation of Non Differentiable Function Using Polynomial 2007 Koo, Lee Feng In any approximation problem, we concerned with a measure of closeness of polynomial ()xp to a function ()xf. Approximations are often obtained by power series expansions in which the higher order terms are dropped. One of the fundamental idea in the differential calculus is that a function can be “locally” approximated by its tangent line. If f be a function defined on an open interval I and let and Ic∈Nn∈. Suppose that the function has n-th derivative at all . Then, the polynomial Ix∈()()ccfTn, is called as Taylor polynomial of order n of function at the point c. If the function is infinitely differentiable on I, so the series is called Taylor Series of f at point c. Taylor Series is one of the fundamental idea in differential calculus. However, Taylor Series only can apply if and only if the function f be differentiated on its interval stated. A differentiable function is a continuous function. But, this is not always true that a continuous function is a differentiable function. Weierstass Approximation Theorem states that every continuous function defined on an interval []ba,can be uniformly approximated as closely as the decided function. Thus, this theorem ensure that every function can be approximate by a polynomial. Consequently, in the research, we develop a new approximation method to approximate the non differentiable function which has singularity at one point, two points and three points by using Fourier series, Lagrange Interpolation and convolution method. We will also discover the asymptotic of Lagrange Interpolation for function where λλ\|\|)(ttf=0>λwith equidistant nodes and discover the convergence of at point t =0. Lastly, we compare the effective of Fourier series and Lagrange method in approximate the non-differentiable function. Interpolation Approximation theory Nondifferentiable functions 2007 Thesis http://psasir.upm.edu.my/id/eprint/5002/ http://psasir.upm.edu.my/id/eprint/5002/1/FS_2007_19.pdf application/pdf en public masters Universiti Putra Malaysia Interpolation Approximation theory Nondifferentiable functions Science
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English
topic	Interpolation Approximation theory Nondifferentiable functions
spellingShingle	Interpolation Approximation theory Nondifferentiable functions Koo, Lee Feng Interpolation and Approximation of Non Differentiable Function Using Polynomial
description	In any approximation problem, we concerned with a measure of closeness of polynomial ()xp to a function ()xf. Approximations are often obtained by power series expansions in which the higher order terms are dropped. One of the fundamental idea in the differential calculus is that a function can be “locally” approximated by its tangent line. If f be a function defined on an open interval I and let and Ic∈Nn∈. Suppose that the function has n-th derivative at all . Then, the polynomial Ix∈()()ccfTn, is called as Taylor polynomial of order n of function at the point c. If the function is infinitely differentiable on I, so the series is called Taylor Series of f at point c. Taylor Series is one of the fundamental idea in differential calculus. However, Taylor Series only can apply if and only if the function f be differentiated on its interval stated. A differentiable function is a continuous function. But, this is not always true that a continuous function is a differentiable function. Weierstass Approximation Theorem states that every continuous function defined on an interval []ba,can be uniformly approximated as closely as the decided function. Thus, this theorem ensure that every function can be approximate by a polynomial. Consequently, in the research, we develop a new approximation method to approximate the non differentiable function which has singularity at one point, two points and three points by using Fourier series, Lagrange Interpolation and convolution method. We will also discover the asymptotic of Lagrange Interpolation for function where λλ\|\|)(ttf=0>λwith equidistant nodes and discover the convergence of at point t =0. Lastly, we compare the effective of Fourier series and Lagrange method in approximate the non-differentiable function.
format	Thesis
qualification_level	Master's degree
author	Koo, Lee Feng
author_facet	Koo, Lee Feng
author_sort	Koo, Lee Feng
title	Interpolation and Approximation of Non Differentiable Function Using Polynomial
title_short	Interpolation and Approximation of Non Differentiable Function Using Polynomial
title_full	Interpolation and Approximation of Non Differentiable Function Using Polynomial
title_fullStr	Interpolation and Approximation of Non Differentiable Function Using Polynomial
title_full_unstemmed	Interpolation and Approximation of Non Differentiable Function Using Polynomial
title_sort	interpolation and approximation of non differentiable function using polynomial
granting_institution	Universiti Putra Malaysia
granting_department	Science
publishDate	2007
url	http://psasir.upm.edu.my/id/eprint/5002/1/FS_2007_19.pdf
_version_	1747810328263000064

Interpolation and Approximation of Non Differentiable Function Using Polynomial

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