Math 172: Tempered Distributions and the Fourier Transform

We have seen that the Fourier transform is well-behaved in the framework of Schwartz functions as well as L, while L is much more awkward. Tempered distributions, which include L, provide a larger framework in which the Fourier transform is well-behaved, and they provide the additional benefit that one can differentiate them arbitrarily many times! To see how this is built up, we start with a reasonable class of objects, such as bounded continuous functions on R, and embed them into a bigger space by a map ι. The bigger space is that of tempered distributions, which we soon define. The idea is that ι is a one-to-one map, thus we may think of bounded continuous functions as tempered distributions by identifying f ∈ C ∞(R) with ι(f). (Below we write ι(f) = ιf often, by analogy of the notation of a sequence, as a distribution itself will be a map, or function, on functions, so we need to write expressions like ιf (φ), which is nicer than (ι(f))(φ).) An analogy is that letters of the English alphabet can be considered as numbers via their ASCII encoding; there are more ASCII codes than letters, but to every letter corresponds a unique ASCII code. One can just think of letters then as numbers, e.g. one thinks of the letter ‘A’ as the decimal number 65, i.e. the letters are thought of as a subset of the integers 0 through 255. So on to distributions. Suppose V is a vector space over F = R or F = C. The algebraic dual of V is the vector space L(V,F) consisting of linear functionals from V to F. That is elements of f ∈ L(V,F) are linear maps f : V → F satisfying f(v + w) = f(v) + f(w), f(cv) = cf(v), v, w ∈ V, c ∈ F.