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Carleman matrix

The Carleman matrix of an infinitely differentiable function ${\displaystyle f($x)} is defined as:

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

so as to satisfy the (Taylor series) equation:

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

For instance, the computation of ${\displaystyle f($x)} by

${\displaystyle f($x)=\sum _{k=0}^{\infty }M[f]_{1,k}x^{k}.~}

simply amounts to the dot-product of row 1 of ${\displaystyle M[$f]} with a column vector ${\displaystyle \left[1,x,x^{2},x^{3},...\right]^{\tau }}$.

The entries of ${\displaystyle M[$f]} in the next row give the 2nd power of ${\displaystyle f($x)} :

${\displaystyle f($x)^{2}=\sum _{k=0}^{\infty }M[f]_{2,k}x^{k}~,}

and also, in order to have the zeroth power of ${\displaystyle f($x)} in${\displaystyle M[$f]} , we adopt the row 0 containing zeros everywhere except the first position, such that

${\displaystyle f($x)^{0}=1=\sum _{k=0}^{\infty }M[f]_{0,k}x^{k}=1+\sum _{k=1}^{\infty }0\cdot x^{k}~.}

Thus, the dot productof${\displaystyle M[$f]} with the column vector ${\displaystyle {\begin{bmatrix}1,x,x^{2},...\end{bmatrix}}^{T}}$ yields the column vector ${\displaystyle \left[1,f($x),f(x)^{2},...\right]^{T}} , i.e.,

${\displaystyle M[$f]{\begin{bmatrix}1\\x\\x^{2}\\x^{3}\\\vdots \end{bmatrix}}={\begin{bmatrix}1\\f(x)\\(f(x))^{2}\\(f(x))^{3}\\\vdots \end{bmatrix}}.}

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}]}

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

${\displaystyle {\begin{aligned}\sum _{n}G[g\circ f]_{ln}\psi _{n}=\psi _{l}\circ (g\circ f)&=(\psi _{l}\circ g)\circ f\\&=\sum _{m}G[$g]_{lm}(\psi _{m}\circ f)\\&=\sum _{m}G[g]_{lm}\sum _{n}G[f]_{mn}\psi _{n}\\&=\sum _{n,m}G[g]_{lm}G[f]_{mn}\psi _{n}\\&=\sum _{n}(\sum _{m}G[g]_{lm}G[f]_{mn})\psi _{n}\end{aligned}}}
  

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]}

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 seriesof${\displaystyle h}$. 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}}{dx^{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)

If${\displaystyle \{e_{n}($x)\}_{n}} is an orthonormal basis for a Hilbert Space with a defined inner product ${\displaystyle \langle f,g\rangle }$, we can set ${\displaystyle \psi _{n}=e_{n}}$ and ${\displaystyle c_{n}($h)} will be ${\displaystyle {\displaystyle \langle h,e_{n}\rangle }}$. Then ${\displaystyle G[$f]_{mn}=c_{n}(e_{m}\circ f)=\langle e_{m}\circ f,e_{n}\rangle } .

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)=\langle h,e_{n}\rangle ={\cfrac {1}{2\pi }}\int _{-\pi }^{\pi }\displaystyle h(x)\cdot e^{-inx}dx} .

Then, therefore, ${\displaystyle {\hat {G}}[$f]_{mn}={\hat {c_{n}}}(e_{m}\circ f)=\langle e_{m}\circ f,e_{n}\rangle } which is

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

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).

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 \\\vdots &\vdots &\vdots &\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 \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}

The Carleman matrix of a constant addition is:

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

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

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

${\displaystyle M_{x}[1+x]_{jk}={\binom {j}{k}}}$

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

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

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

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

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

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

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 \\\vdots &\vdots &\vdots &\vdots &\vdots &\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 \\\vdots &\vdots &\vdots &\vdots &\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 \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}

The Carleman matrix of a linear function is:

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

The Carleman matrix of a function ${\displaystyle f($x)=\sum _{k=1}^{\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 \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}

The Carleman matrix of a function ${\displaystyle f($x)=\sum _{k=0}^{\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 \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}

The Bell matrix or the Jabotinsky matrix of a function ${\displaystyle f($x)} is defined as^[1][2][3]

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

so as to satisfy the equation

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

These matrices were developed in 1947 by Eri Jabotinsky to represent convolutions of polynomials.^[4] It is the transpose of the Carleman matrix and satisfy

${\displaystyle B[f\circ g]=B[$g]B[f]~,} which makes the Bell matrix Bananti-representationof${\displaystyle f($x)} .

• Carleman linearization
• Composition operator
• Function composition
• Schröder's equation
• Bell polynomials

• R Aldrovandi, Special Matrices of Mathematical Physics: Stochastic, Circulant and Bell Matrices, World Scientific, 2001. (preview)
• R. Aldrovandi, L. P. Freitas, Continuous Iteration of Dynamical Maps, online preprint, 1997.
• P. Gralewicz, K. Kowalski, Continuous time evolution from iterated maps and Carleman linearization, online preprint, 2000.
• K Kowalski and W-H Steeb, Nonlinear Dynamical Systems and Carleman Linearization, World Scientific, 1991. (preview)