The Fourier Transformation

Summary

The Fourier transformation of a function u∈L¹ is defined by $${\hat u}(\xi)= \int e^{-i\langle x,\xi\rangle}u(x)dx.$$.

In Section 7.1 we extend the definition to all u∈l’, the space of temperate distributions, which is the smallest subspace of D’, containing L¹ which is invariant under differentiation and multiplication by polynomials. That this is possible is not surprising since the Fourier transformation exchanges differentiation and multiplication by coordinates. (See also the introduction.) It is technically preferable though to define l’, as the dual of the space l of rapidly decreasing test functions. After proving the Fourier inversion formula and basic rules of computation, we study in Section 7.1 the Fourier transforms of L² functions, distributions of compact support, homogeneous distributions and densities on submanifolds. As an application fundamental solutions of elliptic equations are discussed. Section 7.2 is devoted to Poisson’s summation formula and Fourier series expansions. We return to the Fourier-Laplace transform of distributions with compact support in Section 7.3. After proving the Paley-Wiener-Schwartz theorem we give applications such as the existence of fundamental solutions for arbitrary differential operators with constant coefficients, Asgeirsson’s mean value theorem and Kirchoff’s formulas for solutions of the wave equation. The Fourier-Laplace transform of distributions which do not necessarily have compact support is studied in Section 7.4. In particular we compute the Fourier-Laplace transform of the advanced fundamental solution of the wave equation. The Fourier transformation gives a convenient method for approximating C∞ functions by analytic functions. This is used in Section 7.5 to prove the Malgrange preparation theorem after we have recalled the classical analytical counterpart of Weierstrass.

Section 7.6 is devoted to the Fourier transform of Gaussian functions and the convolution operators which they define. This prepares for a rather detailed discussion of the method of stationary phase in Section 7.7, which is a fundamental tool in the study of pseudo-differential and Fourier integral operators in Chapters XVIII and XXV. The Malgrange preparation theorem plays an essential role in many of the proofs. As an application of the simplest form of the method of stationary phase we introduce in Section 7.8 the notion of oscillatory integral. This gives a precise meaning to equations such as $$\delta(\xi)= (2\pi)^{-n}\int e^{i\langle x,\xi\rangle}dx.$$ and will simplify notation later on. In Section 7.9 finally we continue the proof of L^p estimates for convolution operators started in Section 4.5. Applications are given concerning the regularity of solutions of elliptic differential equations with constant coefficients. Although the results are very important in the study of non-linear elliptic differential equations they will not be essential in this book so the reader can skip Section 7.9 without any loss of continuity.