Least squares approximation with constraints

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 \[ {F_e} = \{ f:f( - x) = f(x),f(1) = 0,f \in {L^2}[ - 1,1]\} \] and \[ {F_o} = \{ f:f( - x) = - f(x),f(1) = 0,f \in {L^2}[ - 1,1]\} .\] 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 $\left \| f \right \| = {((f,f))^{1/2}}$, 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 \[ {\phi _{2n}}(x) = \sum \limits _{k = 1}^n {{d_{n,k}}{{(1 - {x^2})}^k}\;{\text {when}} f \in {F_e}} \] and \[ {\phi _{2n + 1}}(x) = x\sum \limits _{k = 1}^n {{e_{n,k}}{{(1 - {x^2})}^k}\;{\text {when}} f \in {F_o}.} \] We apply the general theory to the functions $f(x) = \cos (\pi x/2)$ and $f(x) = {J_0}({a_0}x)$, where ${a_0} = \{ \min x > 0:{J_0}(x) = 0\}$.