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)

Comments

For other normalizations used in defining Fourier transforms, cf. Fourier transform.

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

References

[a1]	K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5
[a2]	D.S. Jones, "The theory of generalized functions" , Cambridge Univ. Press (1982)