Carleman matrix

Generalization

A generalization of the Carleman matrix of a function can be defined around any point, such as:

${\displaystyle M[f{]}_{x_0}=M_x[x-x_0]M[f]M_x[x+x_0]}$

or ${\displaystyle M[f{]}_{x_0}=M[g]}$ where ${\displaystyle g(x)=f(x+x_0)-x_0}$. This allows the matrix power to be related as:

${\displaystyle (M[f{]}_{x_0}{)}^n=M_x[x-x_0]M[f{]}^n{M}_x[x+x_0]}$

General Series

Another way to generalize it even further is think about a general series in the following way:

Let ${\displaystyle h(x)=\sum _n{c}_n(h)\cdot {\psi }_n(x)}$ be a series approximation of ${\displaystyle h(x)}$, where ${\displaystyle \{{\psi }_n(x){\}}_n}$ is a basis of the space containing ${\displaystyle h(x)}$

Assuming that ${\displaystyle \{{\psi }_n(x){\}}_n}$ is also a basis for ${\displaystyle f(x)}$, We can define ${\displaystyle G[f{]}_{mn}=c_n({\psi }_m\circ f)}$, therefore we have ${\displaystyle {\psi }_m\circ f=\sum _n{c}_n({\psi }_m\circ f)\cdot {\psi }_n=\sum _{n}G[f{]}_{mn}\cdot {\psi }_n}$, now we can prove that ${\displaystyle G[g\circ f]=G[g]\cdot G[f]}$, if we assume that ${\displaystyle \{{\psi }_n(x){\}}_n}$ is also a basis for ${\displaystyle g(x)}$ and ${\displaystyle g(f(x))}$.

Let ${\displaystyle g(x)}$ be such that ${\displaystyle {\psi }_l\circ g=\sum _{m}G[g{]}_{lm}\cdot {\psi }_m}$ where ${\displaystyle G[g{]}_{lm}=c_m({\psi }_l\circ g)}$.

Now

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Comparing the first and the last term, and from ${\displaystyle \{{\psi }_n(x){\}}_n}$ being a base for ${\displaystyle f(x)}$, ${\displaystyle g(x)}$ and ${\displaystyle g(f(x))}$ it follows that ${\displaystyle G[g\circ f]=\sum _{m}G[g{]}_{lm}G[f{]}_{mn}=G[g]\cdot G[f]}$

Examples

Rederive (Taylor) Carleman Matrix

If we set ${\displaystyle {\psi }_n(x)=x^n}$ we have the Carleman matrix. Because
${\displaystyle h(x)=\sum _n{c}_n(h)\cdot {\psi }_n(x)=\sum _n{c}_n(h)\cdot x^n}$
then we know that the n-th coefficient ${\displaystyle c_n(h)}$ must be the nth-coefficient of the taylor series of ${\displaystyle h}$. Therefore ${\displaystyle c_n(h)=\frac{1}{n!}{h}^{(n)}(0)}$
Therefore ${\displaystyle G[f{]}_{mn}=c_n({\psi }_m\circ f)=c_n(f(x{)}^m)=\frac{1}{n!}{\left[\frac{d^n}{d{x}^n}(f(x){)}^m\right]}_{x=0}}$
Which is the Carleman matrix given above. (It's important to note that this is not an orthornormal basis)

Carleman Matrix For Orthonormal Basis

If ${\displaystyle \{e_n(x){\}}_n}$ is an orthonormal basis for a Hilbert Space with a defined inner product ${\displaystyle ⟨f,g⟩}$, we can set ${\displaystyle {\psi }_n=e_n}$ and ${\displaystyle c_n(h)}$ will be ${\displaystyle {\displaystyle ⟨h,e_n⟩}}$. Then ${\displaystyle G[f{]}_{mn}=c_n(e_m\circ f)=⟨e_m\circ f,e_n⟩}$.

Carleman Matrix for Fourier Series

If ${\displaystyle e_n(x)=e^{inx}}$ we have the analogous for Fourier Series. Let ${\displaystyle {\hat{c}}_n}$ and ${\displaystyle \hat{G}}$ represent the carleman coefficient and matrix in the fourier basis. Because the basis is orthogonal, we have.

${\displaystyle {\hat{c}}_n(h)=⟨h,e_n⟩=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{\displaystyle h(x)\cdot e^{-inx}dx}}$.

Then, therefore, ${\displaystyle \hat{G}[f{]}_{mn}=\widehat{c_n}(e_m\circ f)=⟨e_m\circ f,e_n⟩}$ which is

${\displaystyle \hat{G}[f{]}_{mn}=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{\displaystyle e^{imf(x)}\cdot e^{-inx}dx}}$

Properties

Carleman matrices satisfy the fundamental relationship

• ${\displaystyle M[f\circ g]=M[f]M[g],}$

which makes the Carleman matrix M a (direct) representation of ${\displaystyle f(x)}$. Here the term ${\displaystyle f\circ g}$ denotes the composition of functions ${\displaystyle f(g(x))}$.

Other properties include:

• ${\displaystyle M[f^n]=M[f{]}^n}$, where ${\displaystyle f^n}$ is an iterated function and
• ${\displaystyle M[f^{-1}]=M[f{]}^{-1}}$, where ${\displaystyle f^{-1}}$ is the inverse function (if the Carleman matrix is invertible).

Examples

The Carleman matrix of a constant is:

${\displaystyle M[a]=\left(\begin{array}{cccc}1& 0& 0& \cdots \\ a& 0& 0& \cdots \\ a^2& 0& 0& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right)}$

The Carleman matrix of the identity function is:

${\displaystyle M_x[x]=\left(\begin{array}{cccc}1& 0& 0& \cdots \\ 0& 1& 0& \cdots \\ 0& 0& 1& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right)}$

The Carleman matrix of a constant addition is:

${\displaystyle M_x[a+x]=\left(\begin{array}{cccc}1& 0& 0& \cdots \\ a& 1& 0& \cdots \\ a^2& 2a& 1& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right)}$

The Carleman matrix of the successor function is equivalent to the Binomial coefficient:

${\displaystyle M_x[1+x]=\left(\begin{array}{ccccc}1& 0& 0& 0& \cdots \\ 1& 1& 0& 0& \cdots \\ 1& 2& 1& 0& \cdots \\ 1& 3& 3& 1& \cdots \\ ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right)}$

${\displaystyle M_x[1+x{]}_{jk}=(\genfrac{}{}{0ex}{}{j}{k})}$

The Carleman matrix of the logarithm is related to the (signed) Stirling numbers of the first kind scaled by factorials:

${\displaystyle M_x[\log (1+x)]=\left(\begin{array}{cccccc}1& 0& 0& 0& 0& \cdots \\ 0& 1& -\frac{1}{2}& \frac{1}{3}& -\frac{1}{4}& \cdots \\ 0& 0& 1& -1& \frac{11}{12}& \cdots \\ 0& 0& 0& 1& -\frac{3}{2}& \cdots \\ 0& 0& 0& 0& 1& \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right)}$

${\displaystyle M_x[\log (1+x){]}_{jk}=s(k,j)\frac{j!}{k!}}$

The Carleman matrix of the logarithm is related to the (unsigned) Stirling numbers of the first kind scaled by factorials:

${\displaystyle M_x[-\log (1-x)]=\left(\begin{array}{cccccc}1& 0& 0& 0& 0& \cdots \\ 0& 1& \frac{1}{2}& \frac{1}{3}& \frac{1}{4}& \cdots \\ 0& 0& 1& 1& \frac{11}{12}& \cdots \\ 0& 0& 0& 1& \frac{3}{2}& \cdots \\ 0& 0& 0& 0& 1& \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right)}$

${\displaystyle M_x[-\log (1-x){]}_{jk}=|s(k,j)|\frac{j!}{k!}}$

The Carleman matrix of the exponential function is related to the Stirling numbers of the second kind scaled by factorials:

${\displaystyle M_x[\exp (x)-1]=\left(\begin{array}{cccccc}1& 0& 0& 0& 0& \cdots \\ 0& 1& \frac{1}{2}& \frac{1}{6}& \frac{1}{24}& \cdots \\ 0& 0& 1& 1& \frac{7}{12}& \cdots \\ 0& 0& 0& 1& \frac{3}{2}& \cdots \\ 0& 0& 0& 0& 1& \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right)}$

${\displaystyle M_x[\exp (x)-1{]}_{jk}=S(k,j)\frac{j!}{k!}}$

The Carleman matrix of exponential functions is:

${\displaystyle M_x[\exp (ax)]=\left(\begin{array}{ccccc}1& 0& 0& 0& \cdots \\ 1& a& \frac{a^2}{2}& \frac{a^3}{6}& \cdots \\ 1& 2a& 2a^2& \frac{4a^3}{3}& \cdots \\ 1& 3a& \frac{9a^2}{2}& \frac{9a^3}{2}& \cdots \\ ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right)}$

${\displaystyle M_x[\exp (ax){]}_{jk}=\frac{(ja{)}^k}{k!}}$

The Carleman matrix of a constant multiple is:

${\displaystyle M_x[cx]=\left(\begin{array}{cccc}1& 0& 0& \cdots \\ 0& c& 0& \cdots \\ 0& 0& c^2& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right)}$

The Carleman matrix of a linear function is:

${\displaystyle M_x[a+cx]=\left(\begin{array}{cccc}1& 0& 0& \cdots \\ a& c& 0& \cdots \\ a^2& 2ac& c^2& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right)}$

The Carleman matrix of a function ${\displaystyle f(x)=\sum _{k=1}^{\mathrm{\infty }}{f}_k{x}^k}$ is:

${\displaystyle M[f]=\left(\begin{array}{cccc}1& 0& 0& \cdots \\ 0& f_1& f_2& \cdots \\ 0& 0& f_1^2& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right)}$

The Carleman matrix of a function ${\displaystyle f(x)=\sum _{k=0}^{\mathrm{\infty }}{f}_k{x}^k}$ is:

${\displaystyle M[f]=\left(\begin{array}{cccc}1& 0& 0& \cdots \\ f_0& f_1& f_2& \cdots \\ f_0^2& 2f_0{f}_1& f_1^2+2f_0{f}_2& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right)}$

Related matrices

The Bell matrix or the Jabotinsky matrix of a function ${\displaystyle f(x)}$ is defined as

${\displaystyle B[f{]}_{jk}=\frac{1}{j!}{\left[\frac{d^j}{d{x}^j}(f(x){)}^k\right]}_{x=0},}$

so as to satisfy the equation

${\displaystyle (f(x){)}^k=\sum _{j=0}^{\mathrm{\infty }}B[f{]}_{jk}{x}^j,}$

These matrices were developed in 1947 by Eri Jabotinsky to represent convolutions of polynomials. It is the transpose of the Carleman matrix and satisfy

${\displaystyle B[f\circ g]=B[g]B[f],}$ which makes the Bell matrix B an anti-representation of ${\displaystyle f(x)}$.