301 APPENDIX B GRAM-SCHMIDT PROCEDURE OF ORTHOGONALIZATION Let V be a finite dimensional inner product space spanned by basis vector functions {w 1, w 2,, w n }. According to the Gram-Schmidt Process an orthogonal set of basis functions {φ 1, φ 2,, φ n } can be constructed from any basis functions {w 1, w 2,, w n } by following three steps: 1. Initially there is no constraining condition on the first basis element φ 1 therefore we can choose φ 1 = w 1. 2. The second basis vector, orthogonal to the first one, can be constructed by satisfying the following condition: φ 2, φ 1 = 0 (B.1) Further, if we write: φ 2 = w 2 cφ 1 (B.2) then we can determine the following value of unknown scalar constant c by substituting this expression for φ 2 in orthogonality condition, given by equation (B.1): c = w 2, φ 1 φ 1, φ 1 (B.3) 3. Continuing the procedure listed in step 2, we can write φ k as: φ k = w k c 1 φ 1 c 2 φ 2 c k 1 φ k 1 (B.4) where, the unknown constants c 1, c 2,, c k 1 can be determined by satisfying
302 following orthogonality conditions: φ k, φ j = 0 For j = 1, 2,, k 1 (B.5) Since, φ 1, φ 2,, φ k 1 are already orthogonal to each other therefore the scalar constant c j can be written as: c j = w k, φ j φ j, φ j (B.6) Therefore, finally we have following general Gram-Schmidt formula for constructing the orthogonal basis vectors φ 1, φ 2,, φ n : φ k = w k k 1 j=1 w k,φ j φ j,φ j φ j, For k = 1, 2,, n (B.7) To construct the orthogonal polynomials of degree n with respect to weight function, 1 x 2 (3 2 x ) on closed interval [ 1, 1], we need to apply the Gram-Schmidt procedure to non-orthogonal monomial basis 1, x, x 2,, x n. First of all, we compute the general expression for x k, x l : x k, x l = 1 1 x k+l (1 x 2 (3 2 x ))dx = 2 k+l+1 6 k+l+3 + 4 k+l+2 k + l is even 0 k + l is odd (B.8) According to this formula, monomials of odd degree are orthogonal to monomials of even degree. Now, if p 0 (x), p 1 (x), denote the resulting orthogonal polynomials then we can begin the process of Gram-Schmidt orthogonalization by letting: φ 0 (x) = 1 (B.9) According to equation (B.7), the next orthogonal polynomial is φ 1 (x) = x x, p 0 p 0, p 0 p 0(x) = x (B.10)
303 Further, recursively using the Gram-Schmidt formula given by equation (B.7), we can generate the orthogonal polynomials given in Table XII, including the recursive form given for φ n (x). In appendix C, we describe an alternative recurrence relation to generate these orthogonal polynomials.
304 APPENDIX C THREE TERM RECURRENCE RELATION TO GENERATE ORTHOGONAL POLYNOMIALS Let V n be a finite dimensional inner product space spanned by orthogonal basis vector functions {φ 1, φ 2,, φ n }, where φ n represent a polynomial of degree n. Next, since xφ(x) V n+1, therefore, there exist numbers c 0, c 1,, c n+1 such that following is true: n+1 xφ n (x) = c i,n φ i (x) (C.1) i=0 Since, φ 0, φ 1,, φ n are orthogonal to each other with respect to weight function w(x), we see that c k,n = 1 µ 2 k xφ n (x)φ k (x)w(x)dx = 1 µ 2 k xv n, φ k, k = 0, 1,, n + 1 (C.2) Where,.,. denotes the inner product defined by weight function w(x) and µ k = φ k, φ k. Further, notice that for k n 2, xv k (x) V n 1 and hence, c k,n = 0, 0 k n 2 and Eq. (C.1) reduces to: xφ n (x) = c n 1,n φ n 1 (x) + c n,n φ n (x) + c n+1,n φ n+1 (x) (C.3) Now, let us assume that and b n are leading coefficients of basis function φ n (x). Hence, from Eq. (C.1), we get: = c n+1,n +1, b n = c n,n + c n+1,n b n+1 (C.4) Also, substituting for k = n 1 in Eq. (C.2), we get c n 1,n = 1 xφ µ 2 n, φ n 1 = n 1 µ2 n c µ 2 n,n 1 n 1 (C.5)
305 Now, from Eqs. (C.4) and (C.5), we get: c n+1,n = +1, c n,n = b n b n+1 +1, c n 1,n = µ2 n µ 2 n 1 1 (C.6) Now, substituting for various c is from Eq. (C.6) in Eq. (C.3), we get following three term recurrence relation: xφ n (x) = a ( n bn φ n+1 (x) + b ) n+1 φ n (x) + +1 +1 µ2 n µ 2 n 1 1 φ n 1 (x) (C.7) Finally, From Eq. (C.7), it is clear that given a sequence of numbers { } and {b n }, one can construct orthogonal polynomials to given weight function w(x). That means, the orthogonal polynomial φ n (x) is unique up to ormalizing factor. In appendix D, we give a more detailed proof of this statement.
306 APPENDIX D UNIQUENESS OF ORTHOGONAL POLYNOMIALS In this appendix, we prove that orthogonal polynomials which satisfy the orthogonality condition of Eq. (3.28) are unique up to ormalizing factor. Let {φ i (x)} and { φ i (x)} are two sets of polynomials which satisfies the following orthogonality condition: φ i (x), φ j (x) φ i (x), φ j (x) 1 1 1 1 w(x)φ i (x)φ j (x)dx = k i δ ij w(x) φ i (x) φ j (x)dx = k i δ ij (D.1) (D.2) Since, φ n (x) is a polynomial of degree n, therefore, we can write it as a linear combination of polynomials {φ 0, φ 1,, φ n } as: φ n (x) = n c i,n φ i (x) i=1 (D.3) Note, by Eq. (D.1) c i,n = 0 for k < n and therefore, φ(x) and φ(x) should be proportional to each other. However, if the leading coefficient of the polynomial φ n (x) is constrained to be one then it is apparent that φ n (x) = φ n (x).