Carleman matrix

In mathematics, a Carleman matrix is a matrix used to convert function composition into matrix multiplication. It is often used in iteration theory to find the continuous iteration of functions which cannot be iterated by pattern recognition alone. Other uses of Carleman matrices occur in the theory of probability generating functions, and Markov chains.
Definition.
The Carleman matrix of an infinitely differentiable function formula_1 is defined as:
so as to satisfy the (Taylor series) equation:
For instance, the computation of formula_1 by
simply amounts to the dot-product of row 1 of formula_6 with a column vector formula_7.
The entries of formula_8 in the next row give the 2nd power of formula_1:
and also, in order to have the zero'th power of formula_1 in formula_8, we adopt the row 0 containing zeros everywhere except the first position, such that
Thus, the dot product of formula_8 with the column vector formula_15 yields the column vector formula_16
Bell matrix.
The Bell matrix of a function formula_1 is defined as
so as to satisfy the equation
so it is the transpose of the above Carleman matrix.
Jabotinsky matrix.
Eri Jabotinsky developed that concept of matrices 1947 for the purpose of representation of convolutions of polynomials. In 1963 he introduces the term "representation matrix", and generalized that concept to two-way-infinite matrices. In that article only functions of the type formula_21 are discussed, but considered for positive *and* negative powers of the function. Several authors refer to the Bell matrices as "Jabotinsky matrix" since (D. Knuth 1992, W.D. Lang 2000), and possibly this shall grow to a more canonical name.
Generalization.
A generalization of the Carleman matrix of a function can be defined around any point, such as:
or formula_23 where formula_24. This allows the matrix power to be related as:
General Series.
Examples.
If we set formula_45 we have the Carleman matrix
If formula_46 is an ortonormal basis for a Hilbert Space with a defined inner product formula_47, we can set formula_48 and formula_49 will be formula_50. If formula_51 we have the analogous for Fourier Series, namely formula_52
Matrix properties.
These matrices satisfy the fundamental relationships:
which makes the Carleman matrix "M" a (direct) representation of formula_1, and the Bell matrix "B" an "anti-representation" of formula_1. Here the term formula_57 denotes the composition of functions formula_58.
Other properties include:
Examples.
The Carleman matrix of a constant is:
The Carleman matrix of the identity function is:
The Carleman matrix of a constant addition is:
The Carleman matrix of the successor function is equivalent to the Binomial coefficient:
The Carleman matrix of the logarithm is related to the (signed) Stirling numbers of the first kind scaled by factorials:
The Carleman matrix of the logarithm is related to the (unsigned) Stirling numbers of the first kind scaled by factorials:
The Carleman matrix of the exponential function is related to the Stirling numbers of the second kind scaled by factorials:
The Carleman matrix of exponential functions is:
The Carleman matrix of a constant multiple is:
The Carleman matrix of a linear function is:
The Carleman matrix of a function formula_78 is:
The Carleman matrix of a function formula_80 is:
Carleman Approximation.
Consider the following autonomous nonlinear system:
where formula_83 denotes the system state vector. Also, formula_84 and formula_85's are known analytic vector functions, and formula_86 is the formula_87 element of an unknown disturbance to the system.
At the desired nominal point, the nonlinear functions in the above system can be approximated by Taylor expansion
where 
formula_89 is the formula_90 partial derivative of formula_1 with respect to formula_92 at formula_93 and formula_94 denotes the formula_90 Kronecker product.
Without loss of generality, we assume that formula_96 is at the origin.
Applying Taylor approximation to the system, we obtain
where formula_98 and formula_99.
Consequently, the following linear system for higher orders of the original states are obtained:
where
formula_101, and similarly formula_102.
Employing Kronecker product operator, the approximated system is presented in the following form
where
x^T &x^