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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Least squares approximation with constraints


Authors: Gradimir V. Milovanović and Staffan Wrigge
Journal: Math. Comp. 46 (1986), 551-565
MSC: Primary 65D15; Secondary 41A30
Corrigendum: Math. Comp. 48 (1987), 854.
MathSciNet review: 829625
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Abstract: In this paper we study two families of functions $ {F_e}$ and $ {F_o}$, and show how to approximate the functions in the interval $ [ - 1,1]$. The functions are assumed to be real when the argument is real. We define

$\displaystyle {F_e} = \{ f:f( - x) = f(x),f(1) = 0,f \in {L^2}[ - 1,1]\} $

and

$\displaystyle {F_o} = \{ f:f( - x) = - f(x),f(1) = 0,f \in {L^2}[ - 1,1]\} .$

Let further $ {\mathcal{P}_m}$ be the set of all real polynomials of degree not higher than m such that the polynomials belong to the set $ {F_e}$ if m is even and to the set $ {F_o}$ if m is odd.

We determine the least squares approximation for the function $ f \in {F_e}$ (or $ {F_o}$) in the class $ {\mathcal{P}_{2n}}$ (or $ {\mathcal{P}_{2n + 1}}$), with respect to the norm $ \left\Vert f \right\Vert = {((f,f))^{1/2}}$, where the inner product is defined by $ (f,g) = \smallint _{ - 1}^1f(x)g(x)w(x)dx$, with $ f,g \in {L^2}[ - 1,1] = L_w^2[ - 1,1]$ and $ w(x) = {(1 - {x^2})^{\lambda - 1/2}}$.

We also consider the general case when f is neither an even nor an odd function but $ f \in {L^2}[ - 1,1]$ and $ f( - 1) = f(1) = 0$.

Using the theory of Gegenbauer polynomials we obtain the approximating polynomials in the form

$\displaystyle {\phi _{2n}}(x) = \sum\limits_{k = 1}^n {{d_{n,k}}{{(1 - {x^2})}^k}\;{\text{when}}\,f \in {F_e}} $

and

$\displaystyle {\phi _{2n + 1}}(x) = x\sum\limits_{k = 1}^n {{e_{n,k}}{{(1 - {x^2})}^k}\;{\text{when}}\,f \in {F_o}.} $

We apply the general theory to the functions $ f(x) = \cos (\pi x/2)$ and $ f(x) = {J_0}({a_0}x)$, where $ {a_0} = \{ \min x > 0:{J_0}(x) = 0\} $.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1986-0829625-7
PII: S 0025-5718(1986)0829625-7
Keywords: Approximation theory, Gegenbauer polynomials
Article copyright: © Copyright 1986 American Mathematical Society