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A Dirac symbol can be interpreted as a linear functional, i.e. a linear mapping from a function space, consisting e.g. of certain real functions, to (or ), having the property
One may think this as the inner product
of a function and another "function" , when the well-known formula
is true.\, Applying this to\, ,\, one gets
i.e. the Laplace transform
By the delay theorem, this result may be generalised to
\\
When introducing a so-called "Dirac delta function", for example
as an "approximation" of Dirac delta, we obtain the Laplace transform
As the Taylor expansion shows, we then have
according to ref.(2).
The Dirac delta , , can be correctly defined as a linear functional, i.e. a linear mapping from a function space, consisting e.g. of certain real functions, to (or ), having the property
One may think of this as an inner product
of a function and another "function" , when the well-known formula
holds.\, By applying this to \, ,\, one gets
i.e. the Laplace transform
By the delay theorem, this result may be generalised to:
[1]
[2]
[3]
- ↑
Schwartz, L. (1950--1951), Th\'eorie des distributions, vols. 1--2, Hermann: Paris.
- ↑
W. Rudin, Functional Analysis ,
McGraw-Hill Book Company, 1973.
- ↑
L. H\"ormander, {\em The Analysis of Linear Partial Differential Operators I,
(Distribution theory and Fourier Analysis)}, 2nd ed, Springer-Verlag, 1990.