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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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
We claim that the sequence \(M=(M_k)\) defined by \(M_k:= \exp (\varphi (k))/k!\) satisfies:
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(1)
\(M\) is a weight sequence, i.e., \(M\) is weakly log-convex,
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(2)
\(\sup _k (\frac{M_{k+1}}{M_k})^{1/k}< \infty \),
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(3)
\(M^{1/k}_k \rightarrow \infty \),
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(4)
\(M_k^{1/k}\) is not almost increasing,
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(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 \(i\ge 2\),
and \(a_{2i-1} \le a_{2i} \le a_{2i+1}\). This proves (1). Let us check (2). Since
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
This implies (3). To show (4) let \(j\) be even. Then
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
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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DOI: https://doi.org/10.1007/s13398-014-0216-0