(1.1) |〈u,v〉| ≤ ‖u‖ ‖v‖. An outline of a proof of this fundamental inequality, when V = R and ‖ · ‖ is the standard Eucledian norm, is given in Exercise 24 of Section 2.7 of [1]. We will give a proof in the general case at the end of this section. LetW be an n dimensional subspace of V and let P : V 7→W be the corresponding projection operator, i.e. if v ∈ V then w = Pv ∈W is the element in W w...

for some fixed τ , which is called the period of f . Though function approximation using orthogonal polynomials is very convenient, there is only one kind of periodic polynomial, that is, a constant. So, polynomials are not good for approximating periodic functions. In this case, trigonometric functions are quite useful. A large class of important computational problems falls under the category...

Very broadly speaking, the Fourier transform is a systematic way to decompose “generic” functions into a superposition of “symmetric” functions. These symmetric functions are usually quite explicit (such as a trigonometric function sin(nx) or cos(nx)), and are often associated with physical concepts such as frequency or energy. What “symmetric” means here will be left vague, but it will usually...

Here are some facts about Fourier Series — useful for pde and elsewhere. Proofs of Lemmas are easy exercises, and not given. On the other hand, proofs of LEMMAS are harder; their proofs are indicated, or a reference is given.