Question

Let V be the space of all even functions on the segment [-1, 1] with the inner product (f(x), g(x)) = ['* f(x)g(x) dx. and let V2 CV be the span of the even polynomials 1 and x². A. Use the Gram-Schmidt process to create an orthonormal basis in V2. B. Find the best approximation of the even function f(x) p(x) = = a + bx². = x by a polynomial C. Find the error of this approximation defined as the norm of the difference: Err=||f(x) - p(x)||.

Answer 1

To create an orthonormal basis in V2 using the Gram-Schmidt process, normalize the first vector and compute the second vector by subtracting the projection of x² onto the first vector. The best approximation of the even function f(x) = a + bx² can be found by taking the inner product of f(x) with 1 and x². The error of this approximation can be calculated by finding the norm of f(x) - p(x).

Step-by-step explanation:

To create an orthonormal basis in V2 using the Gram-Schmidt process, start by normalizing the first vector in V2, which is 1. So, the first vector of the orthonormal basis is 1 divided by the norm of 1, which is 1. Next, compute the second vector in V2 by subtracting the projection of x² onto the first vector from x². Normalize this vector to obtain the second vector of the orthonormal basis.

To find the best approximation of the even function f(x) = a + bx² by a polynomial, we can use the orthogonal projection. Since our orthonormal basis consists of 1 and x², we can find the coefficients a and b by taking the inner product of f(x) with 1 and x². This gives us the equations a = (f(x), 1) and b = (f(x), x²).

The error of this approximation is defined as the norm of the difference between f(x) and the approximation p(x). The error can be calculated by finding the norm of f(x) - p(x), which is ||f(x) - p(x)||.