Distributions and Fourier Transform

Abstract

The Riemann theory of integration in a real vector space of finite dimension is sufficient for our purposes except for Section 1.7, where we outline the FourierPlancherel theory. Let V be a real vector space of finite dimension n. Choose a coordinate system x = (x ₁,…,x _n ) and take the measure dx = dx_i… dx,,, in V. A functionf: V → ℂ is calledRiemann integrable, if it is continuous almost everywhere in V, (i.e., except for a zero measure set) and the norm

$${{\left\| f \right\|}_{1}}\dot{ = }\mathop{{\sup }}\limits_{{B,t}} {{\smallint }_{B}}\left| {{{f}_{t}}} \right|dx$$

is finite, whereBdenotes an arbitrary ball in V and${f_t} = f\;if\;\left| f \right| \leqslant t\;and\;{f_t} = 0$otherwise (truncation of f). We set

$$\int_V {f\;dx \doteq \mathop {\lim }\limits_{r,t \to \infty } } \int_{B(r)} {{f_t}dx}$$

(2)

where B (r) is the ball of radius r centred at the origin (cubes can be taken instead of balls etc.). We shall use standard tools of Riemann integration theory including Fubini’s Theorem. The full range Fubini theorem will be stated in Section 1.7 in the framework of Lebesgue integration theory.