Question

Let X = {−1,0,1} and consider the inner product on F(X) defined as follows:

⟨f, g⟩ = f (−1)g(−1) + 2f (0)g(0) + 4f (1)g(1).

Let E be the subspace of all even functions defined on X. Set g(x) = x² + x for every x ∈ X.

Find the orthogonal projection of g onto E.

Answer 1

The orthogonal projection of the function g(x) = x^2 + x onto the subspace of even functions E in the space F(X) with X = {-1,0,1} is h(x) = x^2, as only the even part of g(x) will contribute to the projection.

Step-by-step explanation:

The student has asked about finding the orthogonal projection of a function g(x) onto a subspace of even functions, using a given inner product on the function space F(X). To project g(x)=x^2+x onto E, the subspace of even functions, we must find a function f(x) within E such that g(x) minus this projection is orthogonal to all functions in E with respect to the given inner product.

Even functions satisfy the condition f(-x) = f(x). Hence, the orthogonal projection of g(x) onto E will be a function that is even. Since g(x) is not even, we only take the even part of g(x), which is x^2. We can disregard the x term because it is odd, and the inner product of it with any even function over the domain given will be zero. Therefore, the orthogonal projection of g(x) onto E is the function h(x) = x^2.

In summary, by using the properties of even and odd functions and the defined inner product, we find that the orthogonal projection of g(x) on the subspace of even functions E over the set X = {-1,0,1} is the even function x^2.