A general method of approximation. I

Abstract:

In this paper we study two families of functions, viz. F and H, and show how to approximate the functions considered in the interval [0,1 ]. The functions are assumed to be real when the argument is real. We define \[ F = \{ f;({\text {i}}) f\left ( {\frac {1}{2} + x} \right ) = f\left ( {\frac {1}{2} - x} \right ),({\text {ii}}) f(0) = f(1) = 0,({\text {iii}})\;f(x)\;{\text {is analytic in a sufficiently large neighborhood of}}\;x = 0\}, \] \[ H = \{ h;({\text {j}})\;h\left ( {\frac {1}{2} + x} \right ) = - h\left ( {\frac {1}{2} - x} \right ),({\text {jj}})\;h(0) = h(1) = 0,({\text {jjj}})\;h(x)\;{\text {is analytic in a sufficiently large neighborhood of}}\;x = 0\}. \] The approximations are defined in the interval [0,1 ] by \[ \min \int _0^1 {{{\left ( {f(x) - \sum \limits _{n = 1}^k {{c_{n,k}}{{[x(1 - x)]}^n}} } \right )}^2}{x^q}{{(1 - x)}^q}\;dx} \] and \[ \min \int _0^1 {{{\left ( {h(x) - (1 - 2x)\sum \limits _{n = 1}^k {{c_{n,k}}{{[x(1 - x)]}^n}} } \right )}^2}{x^q}{{(1 - x)}^q}\;dx} ,\] where $q \in \{ 0,1,2, \ldots \}$. The associated matrices are analyzed using the theory of orthogonal polynomials, especially the Jacobi polynomials ${G_n}(p,q,x)$. We apply the general theory to the basic trigonometric functions $\sin (x)$ and $\cos (x)$.