2.5-1 2.5 Generation of Orthonormal Bases •We have seen that orthonormal bases simplify the calculation of coefficients ˆ n x and the calculation of distances and energies from the coefficients. •This section outlines two classical ways to generate an orthonormal basis from an arbitrary set of basis vectors. (Note that the arbitrary basis and the orthonormal basis span the same space.) •We’ll denote the projection of some onto the space spanned by, say, and by () x t1 ( ) v t2 () v t1 2 , 1 ˆ ˆ ( ) () () Nv v n n n P x t x t x v t = = = ∑ where , 1 ˆ − = x G p [ ] * , () () k n k n v t v t dt ∞ −∞ = ∫ G * ( ) () n n p x t v t dt∞ −∞ = ∫ 2.5-1