Distribution (mathematics) - Tempered Distributions and Fourier Transform

Tempered Distributions and Fourier Transform

By using a larger space of test functions, one can define the tempered distributions, a subspace of D'(Rn). These distributions are useful if one studies the Fourier transform in generality: all tempered distributions have a Fourier transform, but not all distributions have one.

The space of test functions employed here, the so-called Schwartz space S(Rn), is the function space of all infinitely differentiable functions that are rapidly decreasing at infinity along with all partial derivatives. Thus φ : Rn → R is in the Schwartz space provided that any derivative of φ, multiplied with any power of |x|, converges towards 0 for |x| → ∞. These functions form a complete topological vector space with a suitably defined family of seminorms. More precisely, let

for α, β multi-indices of size n. Then φ is a Schwartz function if all the values

The family of seminorms p_{α, β} defines a locally convex topology on the Schwartz-space. The seminorms are, in fact, norms on the Schwartz space, since Schwartz functions are smooth. The Schwartz space is metrizable and complete. Because the Fourier transform changes differentiation by xα into multiplication by xα and vice-versa, this symmetry implies that the Fourier transformations of a Schwartz function is also a Schwartz function.

The space of tempered distributions is defined as the (continuous) dual of the Schwartz space. In other words, a distribution F is a tempered distribution if and only if

is true whenever,

holds for all multi-indices α, β.

The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded (or slow-growing) locally integrable functions; all distributions with compact support and all square-integrable functions are tempered distributions. All locally integrable functions ƒ with at most polynomial growth, i.e. such that ƒ(x) = O(|x|r) for some r, are tempered distributions. This includes all functions in Lp(Rn) for p ≥ 1.

The tempered distributions can also be characterized as slowly growing. This characterization is dual to the rapidly falling behaviour, e.g., of the test functions.

To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform F yields then an automorphism of Schwartz function space, and we can define the Fourier transform of the tempered distribution S by (FS)(ψ) = S(Fψ) for every test function ψ. FS is thus again a tempered distribution. The Fourier transform is a continuous, linear, bijective operator from the space of tempered distributions to itself. This operation is compatible with differentiation in the sense that

and also with convolution: if S is a tempered distribution and ψ is a slowly increasing infinitely differentiable function on Rn (meaning that all derivatives of ψ grow at most as fast as polynomials), then Sψ is again a tempered distribution and

is the convolution of FS and Fψ. In particular, the Fourier transform of the unity function is the δ distribution.