1 Basic Idea
The functional calculus refers to a specific process which enables the expression
to make sense as a linear operator in , for certain scalar functions .
At first sight, and for most functions , there is no reason why the above expression should be associated with a particular linear operator.
But, for example, when is a polynomial , the expression
does indeed refer to a linear operator in .
As another example, when is a matrix in or one is sometimes led to the exponential (http://planetmath.org/MatrixExponential) of
Thus, we are applying the scalar exponential function to a matrix.
Note in this last example that is approximated by polynomials (the partial sums of the series). This provides an idea of how to make sense of if can be approximated by polynomials:
If can be approximated by polynomials then one could try to define
But for that one needs to define what “approximated” means and to assure the above limit exists.
2 More abstractly
There is no reason why one should restrict to linear operators in a normed vector space. In this , we can consider instead a unital topological algebra over a field .
There is no definition in mathematics of functional calculus, but the ideas above show that a functional calculus for an element should be something like an homomorphism from some topological algebra of scalar functions to , that satisfied the following :
3 Functional Calculi
There are some functional calculi of . We give a very brief descprition of each one of them (please follows the links for entries with more detailed explanation).
functional calculus -
A case. It is valid for Hermitian matrices . It associates real valued functions on the spectrum of to elements in the algebra generated by .
Functional calculi provide an of constructing new linear operators having specified out of given ones.
|Date of creation||2013-03-22 17:29:40|
|Last modified on||2013-03-22 17:29:40|
|Last modified by||asteroid (17536)|