Question

Let C=C([0,2π];R) be the vector space of continuous realvalued functions defined over [0,2π] and let V N​

be the subspace of C spanned by {sin(nx),cos(nx)} 0≤n≤N​ where N is a nonnegative integer. The space C (and thus also V ) is an inner product space with inner product given by ⟨f,g⟩=∫ 02π f(x)g(x)dx for any f,g∈C.

(a) Prove that the form ⟨⋅,⋅⟩ defined above is an inner product.

(b) Find an orthonormal basis of VN​ with respect to the inner product.

(c) Prove that C=V N ⊕V N⊥
​
. (d) Let p:C→C be the projection map onto V N​ (and vanishing on V N⊥ ). Compute the following projections: p(x),p(cos( 21 x)),p(sin 2 x)),p(cos((N+1)x). For simplicity, assume that N>1.

Answer 1

An inner product must satisfy four main properties, which ⟨⋅,⋅⟩ does. An orthonormal basis for VN is obtained via the Gram-Schmidt process. The projection p of functions onto VN is computed using inner products with the basis elements of VN.

Step-by-step explanation:

When determining if the form ⟨⋅,⋅⟩ is an inner product, one must verify if it satisfies the following four properties: positivity, linearity in its first argument, commutativity of multiplication, and non-degeneracy.

For an orthonormal basis of VN with respect to the given inner product, apply the Gram-Schmidt process to the basis {sin(nx),cos(nx)} for 0≤n≤N, normalizing at each step.

The statement C = VN ⊕ VN⊥ can be proven by showing that every function in C can be represented uniquely as a sum of an element in VN and an element in VN⊥, where VN⊥ is the orthogonal complement of VN.

The projection p of various functions onto VN can be found by computing the inner product of the given function with each basis element of VN and then constructing the linear combination of the basis elements using these inner products.