The next basic tool we’re going to need is the theory of distributions.

**Distributions **

Distributions are extremely useful because they are both fairly general (including both all integrable functions and things like the Dirac delta function) but also allow for operations such as differentiation. So oftentimes we can obtain distribution solutions to differential equations we are interested in.

Actually, we’ll only discuss here tempered distributions. A **tempered distribution** is a linear functional . Clearly the tempered distributions form a vector space ; it is a locally convex space if we endow it with the weak* topology. It must now be seen how distributions generalize functions. So, if for any , then can be made into a distribution

In fact, we could just assume that grows polynomially. In particular, we have an imbedding .

An example of a distribution that is not a function is the Dirac distribution mapping .

**Operations on distributions **

We use to denote the bilinear pairing . By abuse of notation, when , we will write as well; this fits with the imbedding of into .

Using this form, we can extend many of the normal operators on (whose elements are extremely well-behaved) to . Let be a continuous map; suppose it has an *adjoint map* , i.e.

Then for , we can *define* by the same formula

Since is continuous on , it is also continuous on in the weak* topology, as is easily checked.

The basic example of this is a differential operator. Writing , we see the identity

for is immediate from integration by parts. So given a distribution , we may define by . So any multi-index induces an operator on . As a result, we can talk about distribution solutions to differential equations, which will become important in the future.

For example, it is heuristically said that, on , the Heaviside function defined by for , for is the antiderivative of the delta function. This is actually true in the sense of distributions:

The next basic example of an operation on a distribution is multiplication by a function. Clearly if the map , , is self-adjoint, so we define for a distribution by

**Fourier transforms and convolution**

We can also do convolution—i.e., we can convolve a distribution with an element of . I claim that this is actually the *function* . Indeed, where denotes the application to a function of ,

which becomes

since the partial sums of the integral defining converge in to . Note that the function increases at most polynomially at .

The final example we are interested in today is the Fourier transform of a distribution. This is now routine though, in view of the adjoint property proved the previous time; we just have to define via

(The reversal from to is because the bilinear form we’ve defined is complex-linear; it’s not the hermitian inner product on , so there’s a bit of a change.)

Because the Fourier transform is an isomorphism on , it is one on too.

As one instance of this, we can describe the Fourier transform of the delta function. Indeed:

which means that is precisely the function . And vice versa.

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