Hilbert Space Continued

Purdue University

EAS 657

Geophysical Inverse Theory

Robert L. Nowack

Lecture 3a

Vectors in Hilbert Space

Given a basis for V, (v₁,…v_N) and a defined inner product, the Gram Schmidt procedure can be used to construct an orthonormal basis (,… ).

a) Let

b) Find the “orthogonal projection” of v₂ on (drop a perpendicular), then subtract this part from v₂. The angle between and v₂ is obtained from the inner product as

Now, should be parallel to , where

This gives

Then,

and

If this procedure is continued, then

and

CONVERSE: Given a basis v₁,…v_N of a vector space V, one can always find an inner product such that v₁,…v_N are orthonormal. This can be done by modifying the definition of the inner product.

(x,y) = y^TQx

Choose Q such that x,y are orthonormal.

If , i=1,N are orthonormal, then

for j = 1,…,N

since for an orthonormal basis, (,) =

then,

The representation of a vector using an orthonormal basis is called a Generalized Fourier Series, where

with an inner product such that are orthonormal. Note that care must be taken for infinite dimensional spaces.

Now let’s investigate Signal Spaces L₂[a,b]. This is the space of square-integrable functions on the line [a,b]. Thus,

This is an infinite dimensional space with inner product defined as

and an induced norm

Ex) For the set of all periodic signals with period T on L₂[0,T], choose the complex exponentials

where

and k = , … -1, 0, 1, …

The are orthogonal since

Let

then,

Thus are orthogonal on L₂[0,T]. Let,

k = -¥, …-1,0, + 1,…¥

See Luenberger (1968) p. 61-62. These are orthonormal and can also be shown to be complete in L₂[0,T]. Now any function in L₂[0,T] can be expressed as

where

and

This is called a Generalized Fourier Series since we are free to choose any orthonormal basis. For

then,

The are the Fourier coefficients called the Discrete spectrum of f(t).

A Fourier series pair for L₂[0,T] can be written

where .

If we look at the unbounded interval [-, +], the functions are not square-integrable. But if we relax this requirement and look at basis functions , then

is a generalized function called a Delta function, where

and

Then,

where

Thus,

The spectrum has now coalesced into a continuous function, . Thus, the Continuous Fourier transform pair on with the assistance of generalized functions (a relaxation of the square integrable condition) can be written

However, we aren’t limited to complex exponentials!

Ex) Consider discrete signals where

Let the discrete function be written

then,

where u[k] are the coefficients and are the basis functions.

Define the inner product to be,

Now, is the basis orthonormal? Yes, since

Let

then,

Thus,

The individual sample points provide one basis for a discrete time series.

Ex) Continuous signals that are bandlimited in frequency forming a subspace of

The sampling theorem states that one can completely reconstruct a continuous bandlimited signal from a discrete sampling of the signal. This can be written as

where are the discrete samples with a sampling period T. This is exact if the highest frequency in the signal is less than .

The “impulse response” of this system is where is a sinc function.

In terms of a generalized Fourier series, then the basis functions are with inner product .

Since , then the basis is orthonormal. The coefficient can be written

Polynomials

The polynomials t^k are independent, but not orthogonal on L₂[-1,1]

with inner product on the interval [-1, 1].

We can use Gram Schmidt to find an orthonormal basis, then,

where P_k(t) are called Legendre polynomials with

The problem with the polynomials, t^k, is that they are not nearly orthogonal and an orthonormal basis is preferred. Using different inner products, then different orthogonal polynomials can be obtained.

Ex) Modify the inner product on L₂[-1,1]. Choose