Special Functions

Download original file

This is the original TeX manuscript source. Use the paper’s PDF for typeset equations.

Read the typeset paper

Read this chapter with formatted equations

1 \pageheading{DILATION\; /\; GROWTH THEOREM}{The exact order has positive lower type}
2 The single-channel result extends to finite positive feedback banks. Here a
3 bank has weights $h_j>0$ with $\sum_jh_j=1$, contractions
4 $0<q_j,r_j<1$, and normalized solution
5 \[
6 F'(z)=1+\sum_jh_jF(q_jz)F(r_jz),\qquad F(0)=0.
7 \]
8 Write $M(R)=M_F(R)$ and let $q_j^{\rho_j}+r_j^{\rho_j}=1$.
9 Then, with $\rho=\max_j\rho_j$,
10 \[
11 \boxed{0<\liminf_{R\to\infty}\frac{\log M(R)}{R^\rho}
12 \le\limsup_{R\to\infty}\frac{\log M(R)}{R^\rho}<\infty.}
13 \]
14 Thus logarithmic growth is trapped between two positive multiples of
15 $R^\rho$. In particular,
16 \[
17 \lim_{R\to\infty}\frac{\log\log M(R)}{\log R}=\rho.
18 \]
19 The theorem does not assert that $\log M(R)/R^\rho$ has a limit.
20 \fig{08_growth_type}{Finite-radius logarithmic growth and its proved lower
21 comparison. The normalized curves illustrate scale dependence; the plot is
22 not a proof of convergence to a type constant. Analytic truncation envelopes
23 and ordinary high-precision rounding are recorded separately.}
24 \claim{A lower bound at the exact exponent}
25 Choose a maximizing channel $(h,q,r)$ and integrate over a fixed-width
26 terminal interval. Let
27 \[
28 L=1/h,\quad C=\max\left(\frac{qL}{1-q},\frac{rL}{1-r}\right),\quad
29 B=L+\frac{2}{\min(q,r)},\quad c_-=\frac{\log2}{(B+C)^\rho}.
30 \]
31 The result is the explicit all-radius bound
32 \[
33 \log M(R)\ge c_-(R+C)^\rho,\qquad R\ge2.
34 \]
35 Unlike a fixed fractional window, the terminal window loses only an additive
36 radius. The shift $C$ absorbs that loss without changing the exponent.
37 
38 \pageheading{DILATION\; /\; PROOF}{A translated power comparison}
39 Write $H(R)=\log M(R)$. Positivity gives
40 \[
41 M(R)\ge h\int_{R-L}^{R}M(qt)M(rt)\dd t
42 \ge M(q(R-L))M(r(R-L)),\qquad R\ge B.
43 \]
44 The choice of $C$ ensures
45 \[
46 q(R-L)+C\ge q(R+C),\qquad r(R-L)+C\ge r(R+C).
47 \]
48 On the seed interval $[2,B]$, $H(R)\ge\log R\ge\log2\ge c_-(R+C)^\rho$.
49 If this comparison holds at the two child radii, then
50 \[
51 \begin{aligned}
52 H(R)&\ge H(q(R-L))+H(r(R-L))\\
53 &\ge c_-\{[q(R-L)+C]^\rho+[r(R-L)+C]^\rho\}\\
54 &\ge c_-(q^\rho+r^\rho)(R+C)^\rho=c_-(R+C)^\rho.
55 \end{aligned}
56 \]
57 Let $Q_* =\max(q,r)$ and $d=(1-Q_*)B+Q_*L>0$. Each child radius is at
58 most $R-d$ and at least two. Induction on successive intervals of length
59 $d$ propagates the seed comparison to every $R\ge B$. The argument uses
60 monotonicity of positive powers and works also when $\rho<1$.
61 
62 \claim{Explicit upper constants are available too}
63 Set $Q=\max_j\max(q_j,r_j)$, $m=\min_j\min(q_j,r_j)$,
64 $\Delta=\log(1/m)$, and $B_+=\max_j\log(1/(q_jr_j))+\log2$.
65 Choose an integer $K$ with $Q^{-K}\ge3/m$ and put
66 \[
67 V_K=(2^{K+1}-1)\log2+(2^{K+1}-K-2)\log(1/Q),
68 \]
69 \[
70 c_+=e^{-\rho}(V_K+1+\Delta+B_+).
71 \]
72 Then $\log M(R)\le c_+R^\rho-\log R-B_+$ for $R\ge e$.
73 The tangent majorant $M(1)<2$, followed by the integral recurrence on radii
74 $Q^{-k}$, bounds the initial seed by $V_K$. The logarithmic-radius comparison
75 then supplies the displayed upper constant. The full induction is in Appendix A.
76 
77 These constants are constructive and conservative. Their purpose is to
78 establish two-sided growth, not to estimate the optimal type.

Back to source controls

Back to Special Functions