# Tempered distribution

A tempered distribution is an object from distribution theory , a mathematical branch of functional analysis . A tempered distribution is a special case of a distribution . Laurent Schwartz introduced the temperature-controlled distribution space in 1947 in order to be able to integrate the Fourier transform into his distribution theory.

## Schwartz room

In order to be able to define temperature-controlled distributions, the space of rapidly decreasing functions is explained first. Rapidly falling functions are infinitely differentiable and tend to zero at infinity so quickly that they and all their derivatives fall faster than any polynomial function . The set of all these functions is also known as the Schwartz space and is through ${\ displaystyle {\ mathcal {S}} (\ mathbb {R} ^ {n})}$

${\ displaystyle {\ mathcal {S}} (\ mathbb {R} ^ {n}) = \ {\ phi \ in C ^ {\ infty} (\ mathbb {R} ^ {n}) \, | \, \ forall \ alpha \ in \ mathbb {N} _ {0} ^ {n}, \ beta \ in \ mathbb {N} _ {0} ^ {n} \; \ exists C \ geq 0: \; \ sup _ {x \ in \ mathbb {R} ^ {n}} | x ^ {\ alpha} D ^ {\ beta} \ phi (x) | \ leq C \}}$

Are defined. Through the semi-norms

${\ displaystyle \ | f \ | _ {N} = \ sup _ {x \ in \ mathbb {R} ^ {n}} \ max _ {| \ alpha |, \, | \ beta |

the Schwartz space becomes a metrizable locally convex space . The peculiarity of this space is that the Fourier transform is an automorphism on it. In addition, the room is included in all Sobolev rooms . The space of test functions can be constantly in the Schwartz space embed and lies in this tight . ${\ displaystyle {\ mathcal {D}} (\ mathbb {R} ^ {n})}$

## definition

A tempered distribution is a continuous, linear functional on the Schwartz space, i.e. a continuous linear mapping . Since the set of tempered distributions forms the topological dual space of by definition , this space is also noted. Because of this duality, one speaks of the slowly growing distributions in contrast to the rapidly falling functions. ${\ displaystyle {\ mathcal {S}} (\ mathbb {R} ^ {n}) \ to \ mathbb {C}}$${\ displaystyle {\ mathcal {S}} (\ mathbb {R} ^ {n})}$${\ displaystyle {\ mathcal {S}} '(\ mathbb {R} ^ {n})}$

## Examples

• The class of distributions with compact support is a real subset of the space of temperature-controlled distributions. An example of a compact carrier distribution is the delta distribution .
• Dirac comb
• All distributions that are generated by a polynomial function are tempered distributions. So if it is a polynomial function, then it is a continuous functional${\ displaystyle P}$
${\ displaystyle \ phi \ in {\ mathcal {S}} (\ mathbb {R}) \ mapsto \ int _ {\ mathbb {R}} P (x) \ phi (x) \ mathrm {d} x}$
a tempered distribution. In contrast to the delta distribution or the Dirac comb, these distributions are regular distributions .

## Gelfand's space triple

The Schwartz space lies close to the Hilbert space of the square integrable functions . For this reason, inclusion applies to their dual spaces and from Riesz-Fischer's theorem it follows that this leads to inclusion overall ${\ displaystyle {\ mathcal {S}} (\ mathbb {R} ^ {n})}$ ${\ displaystyle {\ mathcal {H}} = L ^ {2} (\ mathbb {R} ^ {n})}$${\ displaystyle (L ^ {2} (\ mathbb {R} ^ {n})) '\ subset {\ mathcal {S}}' (\ mathbb {R} ^ {n})}$${\ displaystyle L ^ {2} (\ mathbb {R} ^ {n}) \ cong (L ^ {2} (\ mathbb {R} ^ {n})) '.}$

${\ displaystyle {\ mathcal {S}} (\ mathbb {R} ^ {n}) \ subset L ^ {2} (\ mathbb {R} ^ {n}) \ hookrightarrow {\ mathcal {S}} '( \ mathbb {R} ^ {n}).}$

Continuous embedding is the normal identification of a function with a distribution. In other words, is the picture ${\ displaystyle i \ colon L ^ {2} (\ mathbb {R} ^ {n}) \ hookrightarrow {\ mathcal {S}} '(\ mathbb {R} ^ {n})}$${\ displaystyle i}$

${\ displaystyle f \ in L ^ {2} (\ mathbb {R} ^ {n}) \ mapsto \ left (\ phi \ in {\ mathcal {S}} (\ mathbb {R} ^ {n}) \ mapsto \ int _ {\ mathbb {R} ^ {n}} f (x) \ phi (x) \ mathrm {d} x \ right)}$.

The pair gives an example of an extended Hilbert space , or the triple an example of a Gelfandian space triple (based on Israel Gelfand ). The Fourier transform is an automorphism in all three spaces . ${\ displaystyle ({\ mathcal {S}} (\ mathbb {R} ^ {n}), L ^ {2} (\ mathbb {R} ^ {n}))}$${\ displaystyle \ left ({\ mathcal {S}} (\ mathbb {R} ^ {n}), L ^ {2} (\ mathbb {R} ^ {n}), {\ mathcal {S}} ' (\ mathbb {R} ^ {n}) \ right)}$

In contrast to the eigenvalues (i.e. the values ​​of the point spectrum), there are no eigenfunctions in for the values in the continuous part of the spectrum of an operator on . However, there can be distributions which instead satisfy the eigenvalue equation in . Further details can be found in Volume III of the books by Gelfand listed under Literature . When applied to quantum mechanics, this means that the space contains, for example, "eigenfunctions" of the position or momentum operator (in the standard representation these are δ-functions or plane waves ), which are not included because the integral over them Square of the amount diverges. ${\ displaystyle \ lambda \ in \ sigma _ {c} \ left (A \ right)}$ ${\ displaystyle A}$${\ displaystyle L ^ {2}}$ ${\ displaystyle \ lambda \ in \ sigma _ {p} \ left (A \ right)}$${\ displaystyle L ^ {2}}$${\ displaystyle T \ in {\ mathcal {S}} '}$${\ displaystyle \ lambda T = AT}$${\ displaystyle {\ mathcal {S}} '}$${\ displaystyle {\ mathcal {S}} '}$${\ displaystyle L ^ {2} (\ mathbb {R} ^ {3})}$

## Fourier transform

### definition

Be a tempered distribution, the Fourier transform is defined by for all${\ displaystyle u \ in {\ mathcal {S}} '(\ mathbb {R} ^ {n})}$ ${\ displaystyle {\ mathcal {F}} (u)}$${\ displaystyle \ phi \ in {\ mathcal {S}} (\ mathbb {R} ^ {n})}$

${\ displaystyle {\ mathcal {F}} (u) (\ phi): = u ({\ mathcal {F}} (\ phi))}$.

In this context, the Fourier transformation on functions is defined by. There is also another convention for the Fourier transform with the prefactor . However, this is not used in this article. ${\ displaystyle \ textstyle {\ mathcal {F}} (\ phi) (\ xi) = \ int _ {\ mathbb {R} ^ {n}} e ^ {- \ mathrm {i} \ langle x, \ xi \ rangle} \ phi (x) \ mathrm {d} x}$${\ displaystyle {\ tfrac {1} {(2 \ pi) ^ {n / 2}}}}$

### properties

The set is equipped with the weak - * topology . Then the Fourier transform is a continuous, bijective mapping onto . The Fourier archetype of is calculated using the formula${\ displaystyle {\ mathcal {S}} '(\ mathbb {R} ^ {n})}$${\ displaystyle {\ mathcal {S}} '(\ mathbb {R} ^ {n})}$${\ displaystyle {\ mathcal {F}} (u)}$
${\ displaystyle u (\ phi) (- x) = {\ frac {1} {(2 \ pi) ^ {n}}} {\ mathcal {F}} ({\ mathcal {F}} (u)) (\ phi) (x).}$

### example

• Be and the delta distribution to the point . The following then applies to the Fourier transformation${\ displaystyle a \ in \ mathbb {R} ^ {n}}$${\ displaystyle \ delta _ {a} \ in S '(\ mathbb {R} ^ {n})}$${\ displaystyle a}$
${\ displaystyle {\ mathcal {F}} (\ delta _ {a}) (\ phi) = \ delta _ {a} ({\ mathcal {F}} (\ phi)) = {\ mathcal {F}} (\ phi) (a) = \ int _ {\ mathbb {R} ^ {n}} e ^ {- \ mathrm {i} ax} \ phi (x) \ mathrm {d} x}$.
So corresponds to the distribution generated by. In the case corresponds to the distribution generated by. If you also use the prefactor for the Fourier transformation , the result of the example is the distribution that is generated by.${\ displaystyle {\ mathcal {F}} (\ delta _ {a})}$${\ displaystyle x \ mapsto e ^ {- \ mathrm {i} ax}}$${\ displaystyle a = 0}$${\ displaystyle {\ mathcal {F}} (\ delta _ {0})}$${\ displaystyle 1}$${\ displaystyle {\ tfrac {1} {(2 \ pi) ^ {n / 2}}}}$${\ displaystyle {\ tfrac {1} {(2 \ pi) ^ {n / 2}}}}$
• Now be the distribution produced by the constant one function. The obvious approach to compute the expression fails because it leads to a not absolutely convergent integral. To solve it you need the above example and a little trick. It applies${\ displaystyle \ textstyle T_ {1} (\ phi) = \ int _ {\ mathbb {R} ^ {n}} 1 \ cdot \ phi (x) \ mathrm {d} x}$${\ displaystyle {\ mathcal {F}} (T_ {1}) (\ phi)}$
${\ displaystyle {\ mathcal {F}} (T_ {1}) (\ phi) = {\ mathcal {F}} ({\ mathcal {F}} (\ delta _ {0})) (\ phi) = \ delta _ {0} ({\ mathcal {F}} ({\ mathcal {F}} (\ phi))) = (2 \ pi) ^ {n} \ delta _ {0} (\ phi) (- x) = (2 \ pi) ^ {n} \ phi (0) = (2 \ pi) ^ {n} \ delta (\ phi)}$.

### Fourier-Laplace transform

In this section, the Fourier transform is only considered for distributions with compact support. Since the Fourier transform has special properties in this context, it is then called the Fourier-Laplace transform. Be thus a distribution with compact support . Then the Laplace-Fourier transform is through ${\ displaystyle u \ in {\ mathcal {E}} '(\ mathbb {R} ^ {n})}$

${\ displaystyle {\ hat {u}} (\ xi): = u (e ^ {- \ mathrm {i} \ langle \ cdot, \ xi \ rangle})}$

Are defined. This is well defined, because one can show that there is a function which is even analytical - i.e. whole - for all . In addition, this definition agrees with the above definition if the distributions have compact beams. The Paley-Wiener theorem characterizes all of the functions that can occur here as Fourier-Laplace transforms . ${\ displaystyle {\ hat {u}}}$${\ displaystyle \ xi \ in \ mathbb {C}}$

## Laplace transform

A Laplace transformation can also be defined for tempered distributions . This looks similar to the Fourier-Laplace transform from the previous section. Let be a tempered distribution with carrier in , then the Laplace transform of is through ${\ displaystyle u \ in {\ mathcal {S}} '}$${\ displaystyle [0, \ infty [}$${\ displaystyle {\ mathcal {L}}}$${\ displaystyle u}$

${\ displaystyle {\ mathcal {L}} (u) (\ xi): = u (e ^ {- \ langle \ cdot, \ xi \ rangle})}$

Are defined. The result of the transformation is again a holomorphic function that is defined for (but can possibly be analytically extended to a larger set). In contrast to the Fourier-Laplace transform, the Laplace transform is also defined for temperature-controlled distributions that do not have a compact support. This is possible because the decay behavior is better than that of the Fourier kernel . ${\ displaystyle {\ mathfrak {Re}} (\ xi)> 0}$${\ displaystyle e ^ {- \ langle x, \ xi \ rangle}}$${\ displaystyle e ^ {- \ mathrm {i} \ langle x, \ xi \ rangle}}$

## literature

• Lars Hörmander : The Analysis of Linear Partial Differential Operators. Volume 1: Distribution Theory and Fourier Analysis. Second edition. Springer-Verlag, Berlin et al. 1990, ISBN 3-540-52345-6 ( basic teaching of mathematical sciences 256).
• Otto Forster , Joachim Wehler: Fourier Transformation and Wavelets (PDF; 575 kB). 2001 (script).
• RJ Beerends, HG ter Morsche, JC van den Berg, EM van de Vrie: Fourier and Laplace transforms. Cambridge University Press, 2003, ISBN 978-0-521-53441-3 .
• Israel Gelfand : Generalized Functions (Distributions). VEB Deutscher Verlag der Wissenschaften, Berlin (East).
• Volume 1: IM Gelfand, GE Schilow : Generalized functions and calculating with them. 1960 ( University books for mathematics 47, );
• Volume 2: IM Gelfand, GE Schilow: Linear topological spaces, spaces of basic functions and generalized functions. 1962 ( university books for mathematics 48);
• Volume 3: IM Gelfand, GE Schilow: Some questions about the theory of differential equations. 1964 ( university books for mathematics 49);
• Volume 4: IM Gelfand, NJ Wilenkin : Some Applications of Harmonic Analysis. Gelfandian space triple. 1964 ( university books for mathematics 50).
• Volume 5: IM Gelfand, MI Graev: Integral geometry and representation theory 1966, Academic Press.
• Klaus-Heinrich Peters: The connection between mathematics and physics using the example of the history of distributions. A historical study of the fundamentals of physics at the border with mathematics, philosophy and art. 2004 (Hamburg, Univ., Diss., 2003), online (PDF; 2.72 MB) .