Infinite compositions of analytic functions

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In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system.

Notation[edit]

There are several notations describing infinite compositions, including the following:

Forward compositions:

 F_{k,n} (z)= f_k \circ f_{k+1} \circ \cdots \circ f_{n-1} \circ f_n(z)

Backward compositions:

G_{k,n}(z) = f_n\circ f_{n-1} \circ \cdots \circ f_{k+1} \circ f_k(z).

Convergence is interpreted as the existence of  \lim_{n\to \infty} F_{1,n}(z) and  \lim_{n\to\infty} G_{1,n}(z).

For convenience, set Fn(z) = F1,n(z) and Gn(z) = G1,n(z).

Contraction theorem[edit]

Many results can be considered extensions of the following result:

Contraction Theorem for Analytic Functions.[1] Let f be analytic in a simply-connected region S and continuous on the closure S of S. Suppose f(S) is a bounded set contained in S. Then

F_n(z)=f\circ f\circ \cdots \circ f(z)\to \alpha,
the attractive fixed point of f in S, for all z in S.

Infinite compositions of contractive functions[edit]

Let {fn} be a sequence of functions analytic on a simply-connected domain S. Suppose there exists a compact set Ω ⊂ S such that for each n, fn(S) ⊂ Ω.

Forward (inner or right) Compositions Theorem. {Fn(z)} converges uniformly on compact subsets of S to a constant function F(z) = λ.[2]

Backward (outer or left) Compositions Theorem. {Gn(z)} converges uniformly on compact subsets of S to γ ∈ Ω if and only if the sequence of fixed points {γn} of the {fn} converge to γ.[3]

Additional theory resulting from investigations based on these two theorems, particularly Forward Compositions Theorem, include location analysis for the limits obtained here [1]. For a different approach to Backward Compositions Theorem, see [2].

Regarding Backward Compositions Theorem, the example f2n(z) = 1/2 and f2n−1(z) = −1/2 for S = {z : |z| < 1} demonstrates the inadequacy of simply requiring contraction into a compact subset, like Forward Compositions Theorem.

Infinite compositions of other functions[edit]

General analytic functions[edit]

Results[4] involving entire functions include the following, as examples. Set

\begin{align}
f_n(z)&=a_n z + c_{n,2}z^2+c_{n,3} z^3+\cdots \\
\rho_n &= \sup_r \left\lbrace \left| c_{n,r} \right|^{\frac{1}{r-1}} \right\rbrace
\end{align}

Then the following results hold:

Theorem E1.[5] If an ≡ 1,

\sum_{n=1}^\infty \rho_n < \infty
then FnF, entire.

Theorem E2.[4] Set εn = |an−1| suppose there exists non-negative δn, M1, M2, R such that the following holds:

\begin{align}
\sum_{n=1}^{\infty} \varepsilon_n &<\infty, \\
\sum_{n=1}^{\infty} \delta_n&<\infty, \\
\prod_{n=1}^{\infty} (1+\delta_n) &<M_1, \\
\prod_{n=1}^{\infty} (1+\varepsilon_n) &< M_2, \\
\rho_n &< \frac{\delta_n}{R M_1 M_2}.
\end{align}
Then Gn(z) → G(z), analytic for |z| < R. Convergence is uniform on compact subsets of {z : |z| < R}.

Theorem GF3.[4] Let {fn} be a sequence of complex functions defined on S = {z : |z| < M}. Suppose there exists a non-negative sequence {βn} such that

C\sum_{n=1}^{\infty}\beta_n<M,
\left|f_n(z)-z \right|<C\beta_n, \qquad z \in S.
Set R=M-C\sum_{n=1}^{\infty}\beta_n>0. Then Gn(z) → G(z) for |z| < R, uniformly on compact subsets.

Theorem GF4.[4] Let fn(z) = z(1+gn(z)), analytic for |z| < R0, with |gn(z)| ≤ Cβn,

\sum_{n=1}^{\infty} \beta_n<\infty.

Choose 0 < r < R0 and define

R=R(r)=\frac{R_0-r}{\prod_{n=1}^{\infty} \left( 1+C\beta_n \right)}.

Then FnF uniformly for |z| ≤ R. Furthermore,

\left| F'(z) \right|\le \prod_{n=1}^{\infty } {\left( 1+\tfrac{R_0}{r}C\beta_n \right)}.

Linear fractional transformations[edit]

Results[4] for compositions of linear fractional (Möbius) transformations include the following, as examples:

Theorem LFT1. On the set of convergence of a sequence {Fn} of non-singular LFTs, the limit function is either

  • (a) a non-singular LFT,
  • (b) a function taking on two distinct values, or
  • (c) a constant.
In (a), the sequence converges everywhere in the extended plane. In (b), the sequence converges either everywhere, and to the same value everywhere except at one point, or it converges at only two points. Case (c) can occur with every possible set of convergence.[6]

Theorem LFT2. If {Fn} converges to an LFT , then fn converge to the identity function f(z) = z.[7]

Theorem LFT3. If fnf and all functions are hyperbolic or loxodromic Möbius transformations, then Fn(z) → λ, a constant, for all z\ne \beta = \lim_{n\to \infty} \beta_n, where {βn} are the repulsive fixed points of the {fn}.[8]

Theorem LFT4. If fnf where f is parabolic with fixed point γ. Let the fixed-points of the {fn} be {γn} and {βn}. If

\begin{align}
\sum_{n=1}^{\infty}    \left|\gamma_n-\beta_n \right|  &<\infty \\
\sum_{n=1}^{\infty} n \left|\beta_{n+1}-\beta_n \right|&<\infty
\end{align}
then Fn(z) → λ, a constant in the extended complex plane, for all z.[9]

Examples & applications[edit]

Continued fractions[edit]

The value of the infinite continued fraction

\frac{a_1}{b_1+\frac{a_2}{b_2+\ldots}}

may be expressed as the limit of the sequence {Fn(0)} where

f_n(z)=\frac{a_n}{b_n+z}.

As a simple example, a well-known result (Worpitsky Circle*[10]) follows from an application of Theorem (A):

Consider the continued fraction

\frac{a_1\zeta }{1+\frac{a_2\zeta }{1+\ldots}}

with

f_n(z)=\frac{a_n \zeta }{1+z}.

Stipulate that |ζ| < 1 and |z| < R < 1. Then for 0 < r < 1,

|a_n|<rR(1-R)\Rightarrow \left|f_n(z) \right|<rR<R\Rightarrow \frac{a_1\zeta }{1+\frac{a_2\zeta }{1+\ldots}} = F(\zeta ), analytic for |z| < 1.

Set R = 1/2.

Direct functional expansion[edit]

An example illustrating the conversion of a function directly into a composition follows:

Suppose that for |t| > 1, \varphi (tz)=t\left( \varphi (z)+\varphi (z)^2 \right), an entire function with φ(0) = 0, φ′(0) = 1. Then f_n(z)=z+\frac{z^2}{t^n}\Rightarrow F_n(z)\to \varphi (z).[5][11]

Example. f_n(z)=z+\frac{z^2}{2^n}\Rightarrow F_n(z)\to \frac{1}{2}\left( e^{2z}-1 \right)[5]

Calculation of fixed-points[edit]

Theorem (B) can be applied to determine the fixed-points of functions defined by infinite expansions or certain integrals. The following examples illustrate the process:

Example (FP1):[3] For |ζ| ≤ 1 let

G(\zeta )=\frac{ \tfrac{e^{\zeta}}{4}}{3+\zeta +\frac{\tfrac{e^{\zeta}}{8}}{3+\zeta +\frac{\tfrac{e^{\zeta}}{12}}{3+\zeta +\ldots}}}

To find α = G(α), first we define:

\begin{align}
t_n(z)&=\frac{\tfrac{e^{\zeta}}{4n}}{3+\zeta +z} \\
f_n(\zeta )&= t_1\circ t_2\circ \cdots \circ t_n(0)
\end{align}

Then calculate G_n(\zeta )=f_n\circ \cdots \circ f_1(\zeta ) with ζ = 1, which gives: α = 0.087118118... to ten decimal places after ten iterations.

Theorem (FP2).[4] Let φ(ζ, t) be analytic in S = {z : |z| < R} for all t in [0, 1] and continuous in t. Set

f_n (\zeta)=\frac{1}{n}\sum_{k=1}^{n}{\varphi \left( \zeta ,\tfrac{k}{n} \right)}.

If |φ(ζ, t)| ≤ r < R for ζ ∈ S and t ∈ [0, 1], then

\zeta =\int_0^1 \varphi (\zeta ,t)dt
has a unique solution, α in S, with \underset{n\to \infty }{\mathop{\lim }}\,{{G}_{n}}(\zeta )=\alpha .

Evolution functions[edit]

Consider a time interval, normalized to I = [0, 1]. ICAFs can be constructed to describe continuous motion of a point, z, over the interval, but in such a way that at each "instant" the motion is virtually zero (see Zeno's Arrow): For the interval divided into n equal subintervals, 1 ≤ kn set g_{k,n}(z)=z+\varphi_{k,n}(z) analytic - or simply continuous - in a domain S, such that

\lim_{n\to \infty}\varphi_{k,n}(z)=0

for all k and z in S imply g_{k,n}(z)\in S.

Example 1[edit]

g_{k,n}(z)=z+\frac{k}{n^2}f(z).

Now, set T_{1,n}(z)=g_{1,n}(z) and T_{k,n}(z)=g_{k,n}\left(T_{k-1,n}(z) \right). If \lim_{n\to \infty}T_{n,n}(z)=T(z) exists, the initial point z has moved to a new position, T(z), in a fashion described above (for large values of n, g_{k,n}(z)\approx z). It is not difficult to show that f(z) = αz + β, α ≥ 0 implies T_{n,n}(z)\to e^{\frac{\alpha}{2}}z+b\beta . A byproduct of this derivation is the following representation:

\lim_{n\to \infty} \prod_{k=1}^n \left( 1+\frac{2k}{n^2}x \right)=e^x, \qquad x \in \mathbf{R}.

And of course, if f(z) ≡ c, then[12]

T(z)=z+c\int_0^1 tdt.
Two contours flowing towards an attractive fixed point (red on the left). The white contour (c = 2) terminates before reaching the fixed point. The second contour (c(n)=square root of n) terminates at the fixed point. For both contours, n = 10,000

Example 2[edit]

g_n(z)=z+\frac{c_n}{n}\varphi (z),

with f(z) := z + φ(z). Next, set T_{1,n}(z)=g_n(z), T_{k,n}(z)= g_n\left(T_{k-1,n}(z) \right), and Tn(z) = Tn,n(z). Let

T(z)=\lim_{n\to \infty}T_n(z)

when that limit exists. The sequence {Tn(z)} defines contours γ = γ(cn, z) that follow the flow of the vector field f(z). If there exists an attractive fixed point α, meaning |f(z)−α| ≤ ρ|z−α| for 0 ≤ ρ < 1, then Tn(z) → T(z) ≡ α along γ = γ(cn, z), provided (for example) c_n = \sqrt{n}. If cnc > 0, then it seems apparent - though not rigorously proven for many cases[13] - that Tn(z) → T(z), a point on the contour γ = γ(c, z). It is easily seen that

\oint_{\gamma}\varphi (\zeta )d\zeta =\lim_{n\to \infty}\frac{c}{n}\sum_{k=1}^{n}\varphi^2 \left (T_{k-1,n}(z) \right )

and

L(\gamma (z))=\lim_{n\to \infty} \frac{c}{n}\sum_{k=1}^n \left| \varphi \left (T_{k-1,n}(z) \right ) \right|,

when these limits exist.[14]

These concepts are marginally related to active contour theory in image processing.

Self-replicating series & products[edit]

Series[edit]

The series defined recursively by fn(z) = z + gn(z) have the property that the nth term is predicated on the sum of the first n−1 terms. In order to employ theorem (GF3) it is necessary to show boundedness in the following sense: If each fn is defined for |z| < M then |Gn(z)| < M must follow before |fn(z)−z| = |gn(z)| ≤ Cβn is defined for iterative purposes. This is because g_n(G_{n-1}(z)) occurs throughout the expansion. The restriction

|z|<R=M-C\sum_{k=1}^{\infty} \beta_k >0

serves this purpose. Then Gn(z) → G(z) uniformly on the restricted domain.

Example (S1): Set

f_n(z)=z+\frac{1}{\rho n^2}\sqrt{z}, \qquad \rho >\sqrt{\frac{\pi }{6}}

and M = ρ2. Then R = ρ2−(π/6) > 0. Then, if S=\left\{ z: |z|<R,\operatorname{Re}(z)>0 \right\}, z in S implies \left|G_n(z) \right|<M and theorem (GF3) applies, so that

\begin{align}
G_n(z) &=z+g_1(z)+g_2(G_1(z))+g_3(G_2(z))+\cdots + g_n(G_{n-1}(z)) \\ 
&= z+\frac{1}{\rho \cdot 1^2}\sqrt{z}+\frac{1}{\rho \cdot 2^2}\sqrt{G_1(z)}+\frac{1}{\rho \cdot 3^2}\sqrt{G_2(z)}+\cdots +\frac{1}{\rho \cdot n^2} \sqrt{G_{n-1}(z)}
\end{align}

converges absolutely, hence is convergent.

Products[edit]

The product defined recursively by f_n(z)=z\left( 1+g_n(z) \right), |z| ≤ M, have the appearance

G_n(z) = z \prod _{k=1}^n \left( 1+g_k \left( G_{k-1}(z) \right) \right).

In order to apply theorem (GF3) it is required that \left| z\cdot g_n(z) \right|\le C\beta_n where

\sum_{k=1}^{\infty} \beta_k<\infty.

Once again, a boundedness condition must support

\left|G_{n-1}(z)\cdot g_n(G_{n-1}(z))\right|\le C \beta_n.

If one knows Cβn in advance, setting |z| ≤ R = M/P where

\prod_{n=1}^{\infty} \left( 1+C\beta_n\right) =P

suffices. Then Gn(z) → G(z) uniformly on the restricted domain.

Example (P1): Suppose that f_n(z)=z(1+g_n(z)) where g_n(z)=\frac{z^2}{n^3}, observing after a few preliminary computations, that |z| ≤ 1/4 implies |Gn(z)| < 0.27. Then

\left|G_n(z)\cdot \frac{G_n(z)^2}{n^3} \right|<(0.02)\frac{1}{n^3}=C\beta_n

and

G_n(z)=z\cdot \prod_{k=1}^{n-1}\left( 1+\frac{G_k(z)^2}{n^3}\right)

converges uniformly.

References[edit]

  1. ^ P. Henrici, Applied and Computational Complex Analysis, Vol. 1 (Wiley, 1974)
  2. ^ L. Lorentzen, Compositions of contractions, J. Comp & Appl Math. 32 (1990)
  3. ^ a b J. Gill, The use of the sequence Fn(z) = fn o … o f1(z) in computing the fixed points of continued fractions, products, and series, Appl. Numer. Math. 8 (1991)
  4. ^ a b c d e f J. Gill, Convergence of infinite compositions of complex functions, Comm. Anal. Th. Cont. Frac., Vol XIX (2012)
  5. ^ a b c S.Kojima, Convergence of infinite compositions of entire functions, arXiv:1009.2833v1
  6. ^ G. Piranian & W. Thron,Convergence properties of sequences of Linear fractional transformations, Mich. Math. J.,Vol. 4 (1957)
  7. ^ J. DePree & W. Thron,On sequences of Mobius transformations, Math. Zeitschr., Vol. 80 (1962)
  8. ^ A. Magnus & M. Mandell, On convergence of sequences of linear fractional transformations,Math. Zeitschr. 115 (1970)
  9. ^ J. Gill, Infinite compositions of Mobius transformations, Trans. Amer. Math. Soc., Vol176 (1973)
  10. ^ L. Lorentzen, H. Waadeland, Continued Fractions with Applications, North Holland (1992)
  11. ^ N. Steinmetz, Rational Iteration, Walter de Gruyter, Berlin (1993)
  12. ^ J. Gill, Zeno's arrow: A mathematical speculation , Comm. Anal. Th. Cont. Frac., Vol XIX (2012)
  13. ^ J. Gill, John Gill Mathematics Collection, Scribd.com
  14. ^ J. Gill, Progress Report: Zeno Contours in the Complex Plane, Comm. Anal. Th. Cont. Frac., Vol XIX (2012)