2.II.18F

Numerical Analysis | Part IB, 2005

(a) Let {Qn}n0\left\{Q_{n}\right\}_{n \geqslant 0} be a set of polynomials orthogonal with respect to some inner product (,)(\cdot, \cdot) in the interval [a,b][a, b]. Write explicitly the least-squares approximation to fC[a,b]f \in C[a, b] by an nn th-degree polynomial in terms of the polynomials {Qn}n0\left\{Q_{n}\right\}_{n \geqslant 0}.

(b) Let an inner product be defined by the formula

(g,h)=11(1x2)12g(x)h(x)dx(g, h)=\int_{-1}^{1}\left(1-x^{2}\right)^{-\frac{1}{2}} g(x) h(x) d x

Determine the nnth degree polynomial approximation of f(x)=(1x2)12f(x)=\left(1-x^{2}\right)^{\frac{1}{2}} with respect to this inner product as a linear combination of the underlying orthogonal polynomials.

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