Fourier transform of a generalized function

An extension of the Fourier transform from test functions to generalized functions (cf. Generalized function). Let $K$ be a space of test functions on which the Fourier transformation $F$,

$$\phi \rightarrow \ F [ \phi ] = \ \int\limits \phi ( x) e ^ {i ( \xi , x) } \ dx,\ \phi \in K,$$

is defined and on which $F$ is an isomorphism of $K$ onto a space of test functions $\widetilde{K}$. Then the Fourier transformation $f \rightarrow F [ f]$ is defined on the space of generalized functions $\widetilde{K} {} ^ \prime$ by

$$( F [ f], \phi ) = \ ( f, F [ \phi ]),\ \ \phi \in K,$$

and this is an isomorphism of $\widetilde{K} {} ^ \prime$ onto the space of generalized functions $K ^ \prime$.

Examples.

1) $K = S = \widetilde{K}$, $K ^ \prime = S ^ \prime = \widetilde{K} {} ^ \prime$. Here the inverse of $F$ is the operation

$$F ^ {-} 1 [ f] = \ { \frac{1}{( 2 \pi ) ^ {n} } } F [ f (- \xi )],\ \ f \in S ^ \prime ,$$

and the basic formulas for $f \in S ^ \prime$ are

$$D ^ \alpha F [ f] = \ F [( ix) ^ \alpha f],\ \ F [ D ^ \alpha f] = \ (- i \xi ) ^ \alpha F [ f].$$

2) Let $K = \cap _ {s \geq 0 } L _ {2} ^ {s}$, $\widetilde{K} = D _ {L _ {2} } = \cap _ {s \geq 0 } H _ {s}$, $\widetilde{K} {} ^ \prime = D _ {L _ {2} } ^ \prime = \cup _ {s \geq 0 } H _ {-} s$, where $L _ {2} ^ {s}$ is the set of all functions $\phi$ for which $( 1 + ( \xi ) ^ {2} ) ^ {s/2} \phi \in L _ {2}$, and where $H _ {s} = \widetilde{L} {} _ {2} ^ {s}$, $- \infty < s < \infty$.

3) $K = D$, $\widetilde{K} = Z$, where $Z$ is the set of all entire functions $\phi ( z)$ satisfying the growth condition: There is a number $a = a _ \phi \geq 0$ such that for any $N \geq 0$ one can find a $C _ {N} > 0$ such that

$$| \phi ( z) | \leq \ C _ {N} e ^ {a | \mathop{\rm Im} z | } ( 1 + | z |) ^ {-} N ,\ \ z \in \mathbf C ^ {n} .$$

Fourier series of generalized functions.

If a generalized function $f$ is periodic with $n$- period $T = ( T _ {1} \dots T _ {n} )$, $T _ {j} > 0$, then $f \in S ^ \prime$ and it can be expanded in a trigonometric series,

$$f ( x) = \ \sum _ {| k | = 0 } ^ \infty c _ {k} ( f ) e ^ {i ( k \omega , x) } ,\ \ | c _ {k} ( f ) | \leq \ A ( 1 + | k | ) ^ {m} ,$$

converging to $f$ in $S ^ \prime$; here

$$\omega = \left ( \frac{2 \pi }{T _ {1} } \dots \frac{2 \pi }{T _ {n} } \right ) ,\ \ k \omega = \left ( \frac{2 \pi k _ {1} }{T _ {1} } \dots \frac{2 \pi k _ {n} }{T _ {n} } \right ) .$$

Examples.

4) $F ( x ^ \alpha ) = ( 2 \pi ) ^ {n} (- i) ^ {| \alpha | } D ^ \alpha \delta ( \xi )$, in particular $F [ 1] = ( 2 \pi ) ^ {n} \delta ( \xi )$.

5) $F [ D ^ \alpha \delta ] = (- i \xi ) ^ \alpha$, in particular $F [ \delta ] = 1$.

6) $F [ \theta ] = i / ( \xi + i0) = \pi \delta ( \xi ) + iP ( 1/ \xi )$, where $\theta$ is the Heaviside function.

The Fourier transform of the convolution of generalized functions.

Let the direct product $f ( x) \times g ( y)$ of two generalized functions $f$ and $g$ in $D ^ \prime ( \mathbf R ^ {n} )$ admit an extension to functions of the form $\phi ( x + y)$, for all $\phi \in D ( \mathbf R ^ {n} )$. Namely, suppose that for any sequence $\eta _ {k} ( x; y)$, $k \rightarrow \infty$, in $D ( \mathbf R ^ {2n} )$ with the properties: $| D ^ \alpha \eta _ {k} ( x; y) | \leq c _ \alpha$, $\eta _ {k} ( x; y) \rightarrow 1$, $D ^ \alpha \eta _ {k} ( x; y) \rightarrow 0$, $| \alpha | \geq 1$, $k \rightarrow \infty$( uniformly on any compact set), the sequence

$$( f ( x) \times g ( y), \eta _ {k} ( x; y) \phi ( x + y)),\ \ k \rightarrow \infty ,$$

has a limit, denoted by $( f ( x) \times g ( y) , \phi ( x + y))$, which does not depend on the sequence $\{ \eta _ {k} \}$ from the class indicated. In this case the functional $f \star g$ that acts according to the formula $( f \star g, \phi ) = ( f ( x) \times g ( y), \phi ( x + y))$, $\phi \in D ( \mathbf R ^ {n} )$, is called the convolution of the generalized functions $f$ and $g$, $f \star g \in D ^ \prime ( \mathbf R ^ {n} )$. The convolution does not exist for all pairs of generalized functions $f$ and $g$. It automatically exists if for any $R > 0$ the set

$$T _ {R} = \{ { ( x, y) } : { x \in \supp f,\ y \in \supp g,\ | x + y | \leq R } \}$$

is bounded in $\mathbf R ^ {2n}$( in particular if $f$ or $g$ has compact support). If the convolution $f \star g$ exists, then it is commutative: $f \star g = g \star f$; and it commutes with shifts and with derivatives: $f \star D ^ \alpha g = D ^ \alpha ( f \star g) = D ^ \alpha f \star g$; the Dirac $\delta$- function plays the role of "identity" : $f = \delta \star f = f \star \delta$. Convolution is a non-associative operation. However, there are associative (and commutative) convolution algebras. The Dirac delta-function $\delta$ serves as the identity in them. For example, the set $D _ \Gamma ^ \prime$ consisting of generalized functions from $D ^ \prime ( \mathbf R ^ {n} )$ with support in a convex, acute, closed cone $\Gamma$ with vertex at $0$ is a convolution algebra. The set $S _ \Gamma ^ \prime = S ^ \prime \cap D _ \Gamma ^ \prime$ forms a convolution subalgebra of $D _ \Gamma ^ \prime$. Notation: $D _ {+} ^ \prime = D _ {[ 0, \infty ) } ^ \prime$, $S _ {+} ^ \prime = S _ {[ 0, \infty ) } ^ \prime$( when $n = 1$). The formula for the Fourier transform of the convolution

$$F [ f \star g] = F [ f] F [ g]$$

is valid in the following cases:

a) $f \in S ^ \prime$, $g$ has compact support;

b) $f , g \in D _ {L _ {2} } ^ \prime$;

c) $f \in D ^ \prime$, $g$ has compact support;

d) $f , g \in S _ \Gamma ^ \prime$. In this case the product $F [ f] F [ g]$ of the generalized functions $F [ f]$ and $F [ g]$ is understood to be the limit in $S ^ \prime$ of the product $\widetilde{f} ( \zeta ) \widetilde{g} ( \zeta )$, $\zeta = \xi + i \eta$, as $\eta \rightarrow 0$, $\eta \in \mathop{\rm Int} \Gamma ^ {*}$, where $\widetilde{f}$ and $\widetilde{g}$ denote the Laplace transforms of $f$ and $g$( see Generalized functions, product of).

References

 [1] V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1979) (Translated from Russian) [2] I.M. Gel'fand, G.E. Shilov, "Generalized functions" , 1 , Acad. Press (1964) (Translated from Russian) [3] L. Schwartz, "Théorie des distributions" , 2 , Hermann (1951) [4] P. Antosik, J. Mikusiński, R. Sikorski, "Theory of distributions. The sequential approach" , Elsevier (1973) [5] L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983)

The Heaviside function $\theta$ on $\mathbf R$ is defined by $\theta ( x) = 0$ if $x < 0$ and $\theta ( x) = 1$ if $x > 0$.