We start with the polynomials and use Gram-Schmidt. Note that, using integration by parts we get
for .
Let .
Let
To standardize, we let .
Let
Since
we get
To standardize, we let
Let
Since
we get
To standardize, we let
Finally, let
Since
we get
Thus,
To standardize, we divide by its constant term.
Then, are the required polynomials.
Problem 2. The Laguerre polynomials L (x) are orthogonal with respect to the inner product (u,u)=...
Problem 1. Let the inner product (,) be defined by (u.v)xu (x)v (x) dx, and let the norm Iilbe defined by lIul-)Corhe target funtio), and work with the approximating space P4 Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials (x) through degree four. Standardize your polynomials such that p: (1) 1. (a) Form the five-by-five Gram matrix for this inner product with the basis functions p (x) degree 4 approximation o f (x) using the specified norm,...
Problem 1. Let the inner prodct )be deined by (u.v)xu (x) v (x) dx, and let the norm |I-ll be defined by ull , ).Consider the target function f (x) with the approximating space P e', and work 2. Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials p (x) through degree four. Standardize your polynomials such that p, (1) 1 (b) Find the best degree 4 approximation to f(x) using the specified norm, and working with this...
3. Give P an inner product structure by defining (f,9)-of(ag(x)e"da (you do not have to show that this is an inner product). Apply the Gram-Schmidt process to (1, , ) with this inner product. Show your work. Hint: You may want to use the fact that Jooz"e-xdz = nl, which follows from n application of integration by parts. (You do not have to show this.) Fun Fact: The polynomials you find here are the first three Laguerre Polynomials, and are...
6. Let p;(xi = 0,... , n}, with degp;(x) = i, be a set of orthogonal polynomials with respect to the inner product f f(x)g(x) dx. Given a < b, let q(x) be the line mapping a to -1 and b to 1. Prove {p;(q(x))|i = 0,... , n} is a set of orthogonal polynomials with respect to the inner product f(x)g(x) dz, satisfying deg p;(q(x))= i - 6. Let p;(xi = 0,... , n}, with degp;(x) = i, be...
An important fact we have proved is that the family (enr)nez is orthonormal in L (T,C) and complete, in the sense that the Fourier series of f converges to f in the L2-norm. In this exercise, we consider another family possessing these same properties. On [-1, 1], define dn Ln)-1) 0, 1,2, Then Lv is a polynomial of degree n which is called the n-th Legendre polynomial. (a) Show that if f is indefinitely differentiable on [-1,1], thern In particular,...
0 6. 11 points HoltLinAlg2 10.2.012. My Notes Ask Your Teacher Find projsf for Rx)- ex, where S-span1, x and the inner product is eBook 7. 1 points HoltLinAlg2 10.2.014 My Notes Ask Your Teacher Use the Gram-Schmidt process to convert the given set of vectors to an orthogonal basis with respect to the given inner product. (Apply the Gram-Schmidt process in the order the vectors are given and do not normalize.) The set,1,0with respect to the inner product (u,...
question 3 (b) Problem #3: Let R4 have the inner product <u, v>-#1v1 + 2112v2 + 31/3V3 + 414V4 (a) Let w (0, 6, 3,-1). Find |w (b) Let Wbe the subspace spanned by the vectors u (0, 0, 2,1), and u2-,0,,-1) Use the Gram-Schmidt components of the vector v2 into the answer box below, separated with commas process to transform the basis fui. u2 into an orthonormal basis fvi, v23. Enter the Enter your answer symbolically as in these...
NEED (B) AND (C) 2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space C(I-1,1) of continuous real-valued funo- tions on the domain [-1, 1] (b) Use the Gram-Schmidt process to find an orthonormal basis for P2(R) with re- spect to this inner product (c) Find a polynomial q(x) such that for every p E P2R 2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space...
Problem #3: Let R4 have the inner product <u, v> = ulv1 + 2u2v2 + 3u3v3 + 40404 (a) Let w = (0,9,5,-2). Find llwll. (b) Let W be the subspace spanned by the vectors U1 = = (0,0, 2, 1), and u2 = (-3,0,–2, 1). Use the Gram-Schmidt process to transform the basis {uj, u2} into an orthonormal basis {V1, V2}. Enter the components of the vector v2 into the answer box below, separated with commas.
2. Consider R with the weighted inner product = [wn, u, tva, teal"). [ruh, t', talT and w Find the orthogonal projection of w = [1, 2,-1,2]T onto the span of ui-|1,-1, 2, 5]T and u2 [2,1,0,-]. Make sure you are working with an orthonormal basis for u span(u, u2 before you use the usual projection formula. 2. Consider R with the weighted inner product = [wn, u, tva, teal"). [ruh, t', talT and w Find the orthogonal projection of...