Special Functions

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1 \pageheading{DILATION\; /\; TAPS}{A quantitative compact-continuity law}
2 Fix the contractions and let $h,g$ be two normalized nonnegative tap vectors.
3 Set $Q=\max_j\max(q_j,r_j)<1$ and $d=\|h-g\|_1$. Then
4 \[
5 \boxed{\sup_{|z|\le R}|F_h(z)-F_g(z)|
6 \le\frac{d}{4}\,[R f_Q'(R)-f_Q(R)].}
7 \]
8 The right side is an explicitly defined entire majorant. It measures local
9 sensitivity without asserting continuity of the asymptotic growth order.
10 \fig{09_tap_continuity}{Computed weak-tap differences lie below the proved
11 compact bound. For a slow/faster two-channel bank, $d=2\varepsilon$, so the
12 envelope is $\varepsilon[R f_{3/4}'(R)-f_{3/4}(R)]/2$.}
13 Each $c_n(h)$ is a positive polynomial homogeneous of degree $n$ in the
14 weights. Coefficientwise comparison gives
15 $c_n(h)\le d_n(Q)(\sum_jh_j)^n$, where $f_Q=\sum_nd_n(Q)z^{2n+1}$.
16 Every weight partial derivative on the simplex lies in $[0,nd_n(Q)]$.
17 The positive and negative parts of $h-g$ each have mass $d/2$, giving
18 $|c_n(h)-c_n(g)|\le(d/2)nd_n(Q)$.
19 Summing this inequality uses
20 $\sum_nnd_n(Q)R^{2n+1}=[Rf_Q'(R)-f_Q(R)]/2$.
21 
22 \pageheading{DILATION\; /\; PARAMETERS}{An analytic family with a discontinuous invariant}
23 For fixed finite $J$, allow complex $h_j,q_j,r_j$, with
24 $|q_j|,|r_j|<1$. The coefficient recurrence defines polynomials in all these
25 parameters. On a compact parameter set choose $Q<1$ and $W>0$ such that
26 $|q_j|,|r_j|\le Q$ and $\sum_j|h_j|\le W$. Then
27 \[
28 |c_n(h,q,r)|\le W^n d_n(Q),
29 \qquad
30 \sum_n|c_n|R^{2n+1}\le W^{-1/2}f_Q(\sqrt W R).
31 \]
32 Normal convergence proves joint holomorphic dependence on parameters and
33 argument throughout this domain. Signed and complex banks have entire
34 solutions, but the positive-bank growth theorem is not asserted for them.
35 
36 \claim{The weak-tap example now has both sides quantified}
37 For the bank with scales $(1/4,1/4)$ and $(3/4,3/4)$, weights
38 $1-\varepsilon,\varepsilon$, the functions converge on every compact set
39 as $\varepsilon\downarrow0$. The preceding envelope gives a linear bound
40 in $\varepsilon$. At every $\varepsilon>0$, the order is
41 $\log2/\log(4/3)$, while at zero it is $1/2$.
42 
43 The new lower-type theorem does not contradict this: its lower constant
44 depends on the active faster weight and can tend to zero as that channel
45 vanishes. For $\varepsilon=10^{-3}$ one constructive lower constant is
46 about $1.4496\times10^{-9}$; a conservative upper constant is about $90.4392$.
47 
48 \claim{Identifying a bank has unavoidable equivalences}
49 Permuting channels, swapping the scales within a channel, or merging
50 duplicate scale pairs by adding their weights leaves the function unchanged.
51 The two-scale single-channel inverse is exact up to swapping. A general
52 minimal-bank identifiability theorem beyond these equivalences is not proved.
53 
54 The resulting research target is specific: determine optimal growth type,
55 possible logarithmic oscillations, and minimal-bank identifiability. A finite
56 plot cannot settle any of those questions.

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