Talk:Carleman matrix

Categorical interpretation

The section about the categorical interpretation of the Carleman matrix doesn't make sense to me. What would an analytic function on an arbitrary set be? How does one multiply arbitrary infinite matrices (to form the proposed category VecInf)? What is the operation of the supposed Carleman functor on objects? Most important, is there a reference for this to a reliable source? -- Spireguy (talk) 23:42, 13 May 2008 (UTC)

Well, I would imagine the answers would be:

• Analytic functions are not defined on arbitrary sets, only those sets that satisfy the properties of analytic functions.
• Multiplication of finite matrices is Sum A_ij*B_jk, in the case of infinite matrices, the Sum is just from 0 to infinity.
• The operation in SetAn is function composition, the operation in VecInf is matrix multiplication, as stated in the first sentence.
• I have given 4 reliable sources.

Any other questions? AJRobbins (talk) 16:03, 2 March 2009 (UTC)

That still doesn't make sense. I'll be a little blunt, since I think I was too subtle above; my apologies if I'm too direct. First, I'll note that none of the given references even mention the words "category" or "functor"; I checked. So my question about a reliable source still stands. Given how Wikipedia works, that's the crucial thing. However, I highly doubt that such a reliable source exists for the statement currently in the article, because:

1. Since analytic functions aren't defined on arbitrary sets, there is no such category as proposed for SetAn.
2. The phrase "sets that satisfy the properties of analytic functions" doesn't make sense.
3. There is still no operation of the supposed functor on objects. What vector space is supposed to be associated to a given set (or analytic space, or subset of R^n, or whatever)?
4. The proposed sum of A_ij*B_jk will very often diverge, so that isn't a meaningful definition of multiplication of arbitrary infinite matrices, which was what was claimed for VecInf.

Note that I'm not disputing the fact that the operation of taking the Carleman matrix has a functorial feel; I'm saying that the way the article currently tries to define the relevant categories and functors makes no sense, and isn't based on a reliable source. -- Spireguy (talk) 03:35, 3 March 2009 (UTC)

The Carleman-concept only meaningful for functions defined by powerseries

Thinking about the carleman-matrices, and especially the transformation of a function-iteration into matrix-multiplication. If f is, say a zeta-series, how could an iteration be thought? Assume the carleman-matrix for the zeta-function z(s). We may have correctly created the carlemanmatrix Z. Then we have something using derivatives of z(s) the sum of some powers of x by the derivatives of z some function g(x), maybe this is then zeta(x) Them using the next rows gives g(x)^2, g(x)^3,... and we are in the concept of powerseries. In the next iteration, we have then powers of x by powers of g(x) - and I don't belive this is anything related to an iteration of the zeta-series.

Thus - by this sketch I question, if it should be mentioned, that the carleman-concept with its matrix-power-method is meaningful only for function, expressed as polynomials and powerseries in x.

Opinions? Gotti 10:56, 8 June 2008 (UTC)

Yes, it is only meaningful for power series. However, it is not limited to formal power series, but in the case of analytic functions, can be used to find values as well, provided the half-iterate (or whatever iterate you try to find) converges. AJRobbins (talk) 15:49, 2 March 2009 (UTC)

The zeta-function has a power series representation. However, if it is expressed around zero, it has a x^{-1}-term and so we a need "representation-matrix" in the style as Eri Jabotinsky described it in his 1963-artice on "Analytic Iteration" Gotti 08:28, 28 June 2017 (UTC) — Preceding unsigned comment added by Druseltal2005 (talk • contribs)

Relation to Bell-polynomials

The carleman-matrix, or better in its version as Bell-matrix, must be related in some simple way to the Bell-polynomials. As I worked out myself that is simple with the so called "complete Bell-polynomials". But I couldn't settle the point where the "partial bell-polynomials" come into the play. It would be good if either in the article about the Bell-polynomials or here one could (at least) sketch out that relation.

--Gotti 07:23, 1 September 2010 (UTC) —Preceding unsigned comment added by Druseltal2005 (talk • contribs)

References for Jabotinsky matrix

Here are some references for the Jabotinsky matrices; they are taken from D. Knuth, Convolution Polynomials, 1992 (arXiv) - but I have no access the the referred sources so I better proveide that references here so far...

• Jabotinsky, Eri. 1947. “Sur la représentation de la composition de fonctions par un produit de matrices. Applicaton à l’itération de e^x et de e^x-1.” Comptes Rendus Hebdomadaires des Sciences de L’Academie des Sciences, 224: 323-324.

• Jabotinsky, Eri. 1953. “Representation of functions by matrices. Application to Faber polynomials.” Proceedings of the American Mathematical Society 4: 546-553.

• Jabotinsky, Eri. 1963. “Analytic iteration.” Transactions of the American Mathematical Society 108: 457-477.

Gotti 10:54, 15 October 2012 (UTC)

My addition to the section "jabotinsky matrix"

Unfortunately I'm not often editing wikipedia articles, and the editing toolbar (in the new, beta-version, as well in the "source-editing") is killing me. I did not see how I could properly format the formulae, and after that how at all to insert the reference. Sorry, I'm now a bit more tired to learn such things than at 2005 when I first logged in and contributed some material... Gotti 12:18, 19 January 2015 (UTC) — Preceding unsigned comment added by Druseltal2005 (talk • contribs)

Who defined the term "Carleman-matrix" and also their properties as they are?

Title says it all: what is the canonical reference for the introduction of the term "Carleman-matrix"? It was 1963 (if not before) Eri Jabotinsky introduced "representation-matrix". I found much literature using the term "Carleman-linearization" - but that's not enough to root the use of the term "Carleman-matrix"...

Added: The term "carleman embedding matrices" was used by Gralewicz & Kowalski in "Continuous time evolution (...)" in 2000 (see arXiv) Something earlier? — Preceding unsigned comment added by Druseltal2005 (talk • contribs) 07:22, 18 June 2017 (UTC)

--Gotti 07:16, 18 June 2017 (UTC)

Added: from Kowalski,K., (1998): "Nonlinear dynamical systems and (...)"

In 1932 Carleman [3] following ideas of Poincaré and Fredholm demonstrated that nonlinear systems of ordinary differential equations with polynomial nonlinearities can be reduced to an infinite system of linear differential equations. This approach is nowadays referred to as the Carleman linearization or Carleman embedding. (...)

[3] T. Carleman, Acta Mathematica 59, 63 (1932).

--Gotti 07:29, 18 June 2017 (UTC)

(1991) Kowalski & Steeb : "Nonlinear dynamical systems and (...)" pg. 121 "and is called Carleman matrix of order two (...)"

--Gotti 07:40, 18 June 2017 (UTC)

(1985) : N.N.Tarkhanov. "On the Carleman matrix for elliptic systems". Dokl. Acad. Nauk SSSR [Soviet Math.Dokl.], 284, No.2, 294-297 (1985). Gotti 09:59, 20 June 2017 (UTC) — Preceding unsigned comment added by Druseltal2005 (talk • contribs)

Create new article on Carleman linearization and separate it from Carleman matrix

While both topics are related (as far I recall, the matrix obtained in Carleman approximation is the logarithm of a Carleman matrix), I think Carleman linearization could be expanded in a page of its own. Is there any objection to moving the Carleman approximation to a new page and expanding it on this new page? Saung Tadashi (talk) 21:20, 13 January 2021 (UTC)