Fourier Series

Question 1

in a Hilbert space

v is orthogonal to w if…

Answer 1

(v, w) = 0

“inner product”

Question 2

For a set A in a Hilbert space

the set perpendicular to A =

{v | (v, a) = 0 for all a in A}

forms…

“orthogonal complement”

Answer 2

a vector subspace

and

a continuous map x –> (x, a)

in the dual space of the Hilbert space

Question 3

What is an orthogonal basis of H (Hilbert space) ?

Answer 3

1) 0 is not in Basis
2) elements are pairwise orthogonal
3) basis spans a dense subset of H

Question 4

orthonormal set is…

Answer 4

1) parwise orthogonal
2) || v || = 1 for all elements

Question 5

If H (Hilbert space) is separable (has a countable dense subset), then …

Answer 5

H has a finite or countable orthonormal basis​

Question 6

For separable Hilbert space and

an orthonormal basis (e1, e2,…),

what is H_N ?

“the orthogonal projection of v into H_N “

Answer 6

projection: H –> H_N

is defined by

projection(v) = sum to N of (v, ei) ei

Question 7

What does the “Best Approximation Theorem”

say about a separable Hilbert space?

Answer 7

||v - w|| ≥ ||v - projection(v) ||

for w in H_N

“that is the best approx to v in H_N​”

Question 8

Let e1, e2, … be an orthonormal set in H

(a separable Hilbert space)

then what is the i^th Fourier coef?

Answer 8

ê _i (v) = (v, ei)

“inner product”

Question 9

What is Bessel’s inequality?

Answer 9

∑ | ê_i (v) | ≤ || v ||²

Question 10

When is e1, e2, … an orthonormal set

a basis of H

in term so Fourier?

Answer 10

if and only if

∑ | ê_i (v) |² = |v|²​ for all v

Question 11

Riesz-Fischer

Answer 11

Let e1, e2, … be an orthonormal basis of H,

then v —> { ê(v) } i=1 to infinity

is an isometric isomorphism of H with l²​