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Equivalence of stability properties for ultradifferentiable function classes

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

We characterize stability under composition, inversion, and solution of ordinary differential equations for ultradifferentiable classes, and prove that all these stability properties are equivalent.

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Correspondence to Armin Rainer.

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Supported by FWF-Projects P 26735-N25 and P 23028-N13.

Appendix: Weight sequences as required in Remark 2

Appendix: Weight sequences as required in Remark 2

Let us now find explicit sequences that satisfy the requirements of Remark 2. To this end we construct a weight sequence \(M=(M_k)\) such that \((M_{k+1}/M_k)^{1/k}\) is bounded, \(M^{1/k}_k\) tends to \(\infty \) but is not almost increasing, and \(k!^s \le M_k \le k!^t\) for suitable \(s,t > 0\) and sufficiently large \(k\). Since for every Gevrey sequence \(G^s =(k!^s)_k\) (where \(s\ge 0\)), \((G^s_k)^{1/k}\) is increasing (and tends to \(\infty \) if \(s>0\)), the pair of sequences \((G^s, M)\), or \((M,G^t)\), will fulfill the requirements of Remark 2 (after adjusting finitely many terms of one sequence).

Let \(k_j:=2 \uparrow \uparrow j = 2^{2^{{\cdot }^{{\cdot }^ 2}}}\) (\(j\) times) for \(j\ge 1\) and \(k_0:=0\). Let \(\varphi : [0,\infty ) \rightarrow [0,\infty )\) be the function whose graph is the polygon with vertices \(\{v_j=(k_j,\varphi (k_j)): j \in \mathbb {N}\}\) defined by

$$\begin{aligned} \varphi (0):=0, \quad \varphi (2):= 8 \log 2,\quad \varphi (k_j) := {\left\{ \begin{array}{ll} k_j \log k_{j+1} &{} \textit{ j} \text { even} \\ k_j \log (k_j k_{j-2}) &{} \textit{ j} \text { odd} \end{array}\right. }, \quad (j\ge 2). \end{aligned}$$

We claim that the sequence \(M=(M_k)\) defined by \(M_k:= \exp (\varphi (k))/k!\) satisfies:

  1. (1)

    \(M\) is a weight sequence, i.e., \(M\) is weakly log-convex,

  2. (2)

    \(\sup _k (\frac{M_{k+1}}{M_k})^{1/k}< \infty \),

  3. (3)

    \(M^{1/k}_k \rightarrow \infty \),

  4. (4)

    \(M_k^{1/k}\) is not almost increasing,

  5. (5)

    \(k!^s \le M_k \le k!^t\) for all \(0\le s \le 1/4\), \(t> 3\), and all \(k\ge k_0(t)\).

To see that \(M\) is weakly log-convex it suffices to show that the slopes of the line segments in the graph of \(\varphi \) are increasing. Let \(a_j\) denote the slope of the line segment left of the vertex \(v_j\). Then, for \(i\ge 2\),

$$\begin{aligned} a_{2i-1}&= \frac{k_{2i-1} \log (k_{2i-1} k_{2i-3}) - k_{2i-2} \log k_{2i-1}}{k_{2i-1}-k_{2i-2}} = \frac{\frac{5}{4} k_{2i-2}-1}{k_{2i-2}-1} \log k_{2i-1} \nonumber \\ a_{2i}&= \frac{k_{2i} \log k_{2i+1} - k_{2i-1} \log (k_{2i-1} k_{2i-3})}{k_{2i}-k_{2i-1}} = \frac{4 k_{2i-1}-\frac{5}{4}}{k_{2i-1}-1} \log k_{2i-1} \nonumber \\ a_{2i+1}&= \frac{k_{2i+1} \log (k_{2i+1} k_{2i-1}) - k_{2i} \log k_{2i+1}}{k_{2i+1}-k_{2i}} = \frac{5 k_{2i}-4}{k_{2i}-1} \log k_{2i-1} \end{aligned}$$
(A.1)

and \(a_{2i-1} \le a_{2i} \le a_{2i+1}\). This proves (1). Let us check (2). Since

$$\begin{aligned} \Big (\frac{M_{k+1}}{M_k}\Big )^{1/k} = \frac{\exp (\frac{\varphi (k+1)}{k}-\frac{\varphi (k)}{k})}{(k+1)^{1/k}} \le \exp (\tfrac{\varphi (k+1)}{k}-\tfrac{\varphi (k)}{k}), \end{aligned}$$

it suffices to show that \(\tfrac{\varphi (k+1)}{k}-\tfrac{\varphi (k)}{k}\) is bounded or equivalently that the slope of the line segments of \(\varphi \) increases at most linearly in \(k\). This is obvious from (A.1). Thanks to \(k! \le k^k \le e^k k!\) we have

$$\begin{aligned} \frac{\exp (\frac{\varphi (k)}{k})}{k} \le M^{1/k}_k \le \frac{\exp (\frac{\varphi (k)}{k}+1)}{k}. \end{aligned}$$

This implies (3). To show (4) let \(j\) be even. Then

$$\begin{aligned} \log \frac{M^{1/k_j}_{k_j}}{M^{1/k_{j+1}}_{k_{j+1}}} \ge \log k_j + \log k_{j+1} - \log (k_{j+1} k_{j-1}) -1 = \log k_{j-1}-1 \rightarrow \infty , \end{aligned}$$

as required. Finally, \(k!^s \le M_k \le k!^t\) is equivalent to \(1+s \le \tfrac{\varphi (k)}{\log k!} \le 1+t\). We have

$$\begin{aligned} \frac{\varphi (k_j)}{k_j\log k_j} = {\left\{ \begin{array}{ll} 2 &{} \textit{ j} \text { even} \\ 1+\frac{1}{4} &{} \textit{ j} \text { odd} \end{array}\right. } \end{aligned}$$

and \(\tfrac{\varphi (k)}{k\log k} \le \tfrac{\varphi (k)}{\log k!} \le 2 \tfrac{\varphi (k)}{k\log k}\). So the first inequality in (5) follows thanks to the fact that \(k!^s\) is log-convex for each \(s\ge 0\). For the second inequality we observe that in view of (A.1) the slope \(a_i\) is dominated by the increment \(\log ((k+1)!^{1+t}) - \log (k!^{1+t}) = (1+t) \log (k+1)\) for all \(k_{i-1}\le k < k_i\) provided that \(t>3\) and that \(i\) is sufficiently large depending on \(t\).

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Rainer, A., Schindl, G. Equivalence of stability properties for ultradifferentiable function classes. RACSAM 110, 17–32 (2016). https://doi.org/10.1007/s13398-014-0216-0

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