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
(0.1) |
When introducing a so-called “Dirac delta function”, for example
for for |
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
(0.2) |
By the delay theorem, this result may be generalised to:
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