Use the inner product

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Special functions and orthogonal polynomials

Conditions of Validity, Ch Inner products, orthogonality and the adjoint

Problem

Use the inner product ⟨f,g⟩=∫01f∗g dx\langle f,g\rangle=\int_0^1f^*g\,\mathrm{d}x, weight w=1w=1 on [0,1][0,1], on real polynomials. ((a)) Run Gram-Schmidt on 1,x,x21,x,x^2 and find u1,u2,u3u_1,u_2,u_3 with leading coefficient 11. Show that each uku_k is a multiple of the Legendre polynomial Pk−1(2x−1)P_{k-1}(2x-1), and explain from the substitution t=2x−1t=2x-1 why this had to happen. ((b)) Find the best approximation to f=x3f=x^3 from S=span{1,x,x2}\mathcal{S}=\mathrm{span}\{1,x,x^2\} in this norm, as a polynomial in xx. ((c)) Find the residual norm ∥x3−PSx3∥\|x^3-P_{\mathcal S}x^3\|.

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