Some useful Facts on Fourier transformation

We denote by $L ^1 (r^n) $ The space of Lebesgue integrable functions on $R ^n$. For $f \in l^1 (r^n) $, the Fourier transformation $\widehat{f}$ of $F $ are defined by

$$\WIDEHAT{F} (\xi) =\int f (x) E^{i\xi \cdot x}dx, \quad \xi \in r^n.$$

Fact 1. $\WIDEHAT{F} (\XI) $ is uniformly continuous on $R ^n.$

Fact 2. If $h (x) =f (x) \ast g (x): =\int f (x-y) g (y) dy,$ then $\widehat{h} (\xi) =\widehat{f} (\XI) \cdot \widehat{g} (\XI). $

Fact 3. Let $f \in l^1 (r^n) $ and $XF (x) \in l^1 (r^n) $. Then $\widehat (f) $ is differentiable and $\frac{d}{d\xi}\widehat{f} (\XI) =\widehat{(-IXF)} (\XI). $

Fact 4. (Riemann-lebesgue Lemma) For $f \in l^1 (r^n) $, $\lim\limits_{|\xi|\to 0}\widehat{f} (\XI) =0.$

Important Example: Let $f (x) =|x|^{-s}, X\in r^n$. Then $\widehat{f} (\XI) =c (s,n) |x|^{s-n}.$ See "Lectures in Harmonicanalysis" by Thomas H. Wolff.

We denote by $M (r^n) $ The space of all finite Borel measures on $R ^n.$ $M (r^n) $ was identified with the dual space of $C _0 ( R^n) $--the (sup-normed) space of all continuous fnctions in $R ^n$ which vanish at infity--by means of the coupling

$$<\MU, F>=\int fd\mu.\quad f\in c_0 (r^n), \mu\in M (r^n). $$

The convolution of a measure $\mu\in M (r^n) $ and a function $\varphi\in c_0 (r^n) $ is defined by

$$ (\varphi \ast\mu) (x) =\int \varphi (XY) d\mu (y). $$

Clearly, $\varphi\ast\mu\in C_0 (r^n) $ and $\|\varphi\ast\mu\|\le \|\mu\|\|\varphi\|. $

The Fourier transformation of a measure $\mu\in M (r^n) $ is defined by

$$\WIDEHAT{\MU} (\XI) =\int E^{i\xi\cdot x}d\mu (x). $$

Remark: If $\mu \ll \mathcal{l}^n,$ say $d \mu=f (x) dx,$ then $\widehat{\mu} (\xi) =\widehat{f} (\XI). $

Fact 5 (Parseval ' s theorem) Let $\mu \in M (r^n) $ and let $f $ is a continuous function in $L ^1 (r^n) $ such that $\widehat{f}\in l^1 (r^n) $. Then

$$\int f (x) d\mu (x) = (2\PI) ^n\int \widehat{f} (\XI) \widehat{\mu} (-\xi) dx= (2\PI) ^n\int \widehat{f} (\XI) \overline{\ WIDEHAT{\MU} (\XI)}dx.$$

Some useful Facts on Fourier transformation