How big is Hilbert space?

Now we understand this question to mean, what is the dimension of Hilbert space. We realized the importance of having an inner product on ${\cal L}^{2}(E)$, since this allowed us to employ the techniques that were developed in Linear Algebra so simplify our search for a basis. Naturally, our goal was to produce an orthornormal basis. In order to make the problem tractable, we assumed that E = [0,1]. (Out of laziness I have dropped the notation [f] in this section and will simply write f.)

Since we believed that ${\cal L}^{2}$ was a ``big'' vector space, we initially tried to produce an infinite linearly independent set. As our first guess, we chose the functions $f_{i} = \chi_{[0,2^{-i}]}$. We observed

1.

(a)

The collection $\{ f_{i} \}$ is a linearly independent set that does not span ${\cal L}^{2}(E)$.

(b)

It is possible to augment this collection of functions so as to produce a linearly independent set that spans.

2.

Since by definition linear combinations are finite, we introduced a new notation of linear combination. Given a sequence of orthonormal functions $\{ f_{i} \}$, we said we would allow infinite linear combinations

$$\sum_{i=1}^{\infty} a_{i}f_{i}$$

if the infinite sum of the squares of the coefficients of these functions is a convergent series. That is, the sequence of partial sums $s_{n} = \sum_{i=1}^{n} a_{i}f_{i}$ converges if and only if

$$\sum_{i=1}^{\infty} \mid a_{i} \mid^{2} < \infty$$

3.

We said that the sequence of functions $\chi_{[0,1]}, \ \chi_{[0,1/2]}, \ \chi_{[0,1/4]}, \ \chi_{[1/2,3/4]}, \ldots$ is linearly independent and spans ${\cal L}^{2}([0,1])$, but is not orthonormal.

4.

If we begin with the function $f(x) = \chi_{[0,1/2]} - \chi_{[1/2]}$, the Haar function (which resembles the Rademacher functions we used last semester), we can generate an orthogonal spanning set by scaling and shifting f. We used $f(x), \ f(2x), \ f(2(x - 1/2)), \ldots$. These can be normalized to produce an orthormal basis.

The Haar function is called a mother wavelet and the resulting basis is called a wavelet basis. We need to find the expression for the entire basis in terms of f, which means finding the complete pattern for the scaling and shifting coefficients, and we need to find the normalization factors in order to make the basis orthonormal.

The wavelet basis for ${\cal L}^{2}([0,1])$ extends by shifting to produce a basis for ${\cal L}^{2}({\bf R})$.

5.

The set of functions $\{1,\sin(mx),\cos(mx)\}_{m=1\ldots \infty}$ is an orthogonal set of functions in ${\cal L}^{2}([0,2\pi])$. We need to find the normalization factors in order to make the set orthonormal and we need to show that this set spans ${\cal L}^{2}([0,2\pi])$. This collection of functions is the Fourier basis for ${\cal L}^{2}([0,2\pi])$.

How would you adjust this collection of functions to produce a basis for ${\cal L}^{2}([a,b])$? (It takes a bit more than saying expand the domain.) This basis does not extend to a basis for ${\cal L}^{2}({\bf R})$.

6.

(New.) Show that the polynomials $\{x^{n}\}_{n=0\ldots \infty}$are linearly independent in ${\cal L}^{2}([0,1])$, but not in ${\cal L}^{2}({\bf R})$. (The second part is a trick question.) Show that these functions are not orthonormal. Can you produce an orthonormal set in ${\cal L}^{2}([0,1])$, $f_{0}, f_{1}, \ldots$ so that f_i is a polynomial of degree i. Does this set span ${\cal L}^{2}([0,1])$?