Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

Abstract

In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite-dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite-dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite-dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a two-dimensional manifold of equilibria of the Cahn–Hilliard equation.

Appendices

Appendix 1: Convolution estimates

In this Appendix, we provide the necessary convolution estimates required to construct the radii polynomials for the Cahn–Hilliard equation studied in Sect. 5. We decided to include all formulas and proofs so that the paper is self-contained. Note, however, that these analytic convolution estimates are taken directly from [15] for estimates concerning quadratic and cubic nonlinearities.

Consider a decay rate $q \ge 2$, a computational parameter $M \ge 6$ and define, for $k \ge 3$,

$$\begin{aligned} \gamma _k = \gamma _k(q) \,\mathop {=}\limits ^{{{\mathrm{def}}}}\,2 \left[ \frac{k}{k-1}\right] ^q + \left[ \frac{4\ln (k-2)}{k} + \frac{\pi ^2-6}{3} \right] \left[ \frac{2}{k} + \frac{1}{2} \right] ^{q-2}. \end{aligned}$$

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Lemma 6.1

For $q \ge 2$ and $k \ge 4$, we have

$$\begin{aligned} \sum _{k_1 = 1}^{k-1} \frac{k^q}{k_1^q (k-k_1)^q} \le \gamma _k. \end{aligned}$$

Proof

First observe that

$$\begin{aligned} \sum _{k_1 = 1}^{k-1} \frac{k^q}{k_1^q (k-k_1)^q}= & {} 2 {\left[ \frac{k}{k-1} \right] }^q + k^{q-1} \sum _{k_1 = 2}^{k-2} \frac{k}{k_1^q (k-k_1)^q} \\= & {} 2 {\left[ \frac{k}{k-1} \right] }^q + k^{q-1} \left[ \sum _{k_1 = 2}^{k-2} \frac{k-k_1}{k_1^q (k-k_1)^q} + \sum _{k_1 = 2}^{k-2} \frac{k_1}{k_1^q (k-k_1)^q} \right] \\= & {} 2 {\left[ \frac{k}{k-1} \right] }^q + k^{q-1} \left[ \sum _{k_1 = 2}^{k-2} \frac{1}{k_1^q (k-k_1)^{q-1}} + \sum _{k_1 = 2}^{k-2} \frac{1}{k_1^{q-1} (k-k_1)^q} \right] \\= & {} 2 {\left[ \frac{k}{k-1} \right] }^q + 2 \sum _{k_1 = 2}^{k-2} \frac{k^{q-1}}{k_1^{q-1} (k-k_1)^q}. \end{aligned}$$

Using the above, we define

$$\begin{aligned} \phi _k^{(q)} := \sum _{k_1=2}^{k-2} \frac{k^{q-1}}{k_1^{q-1} (k-k_1)^q} = \frac{1}{2} \sum _{k_1=2}^{k-2} \frac{k^q}{k_1^q (k-k_1)^q}. \end{aligned}$$

We then obtain the following recurrence inequality

$$\begin{aligned} \phi _k^{(q)}= & {} \sum _{k_1=2}^{k-2} \frac{k^{q-1}}{k_1^{q-1} (k-k_1)^q} = k^{q-2} \sum _{k_1=2}^{k-2} \frac{(k-k_1) + k_1}{k_1^{q-1} (k-k_1)^q} \\= & {} \frac{1}{k} \sum _{k_1=2}^{k-2} \frac{k^{q-1}}{k_1^{q-1} (k-k_1)^{q-1}} + \sum _{k_1=2}^{k-2} \frac{k^{q-2}}{k_1^{q-2} (k-k_1)^q} \\\le & {} \frac{1}{k} \sum _{k_1=2}^{k-2} \frac{k^{q-1}}{k_1^{q-1} (k-k_1)^{q-1}} + \frac{1}{2} \sum _{k_1=2}^{k-2} \frac{k^{q-2}}{k_1^{q-2} (k-k_1)^{q-1}} \;\; = \;\; \left[ \frac{2}{k} + \frac{1}{2} \right] \phi _k^{(q-1)}. \end{aligned}$$

Applying the above inequality $q-2$ times, we get

$$\begin{aligned} \phi _k^{(q)} \le \phi _k^{(2)} {\left[ \frac{2}{k} + \frac{1}{2} \right] }^{q-2}. \end{aligned}$$

Also

$$\begin{aligned} \phi _k^{(2)}= & {} \sum _{k_1=2}^{k-2} \frac{k}{k_1 (k-k_1)^2} = \sum _{k_1=2}^{k-2} \frac{1}{k_1 (k-k_1)} + \sum _{k_1=2}^{k-2} \frac{1}{(k-k_1)^2} \\= & {} \frac{1}{k} \left[ \sum _{k_1=2}^{k-2} \frac{1}{k_1} + \sum _{k_1=2}^{k-2} \frac{1}{k-k_1} \right] + \sum _{k_1=2}^{k-2} \frac{1}{(k-k_1)^2} \\= & {} \frac{2}{k} \sum _{k_1=2}^{k-2} \frac{1}{k_1} + \sum _{k_1=2}^{k-2} \frac{1}{k_1^2} \le \frac{2}{k} \ln {(k-2)} + \frac{\pi ^2}{6} -1. \end{aligned}$$

Using the above inequalities, we get

$$\begin{aligned} \sum _{k_1 = 1}^{k-1} \frac{k^q}{k_1^q (k-k_1)^q}= & {} 2 {\left[ \frac{k}{k-1} \right] }^q + 2 \phi _k^{(q)} \le 2 {\left[ \frac{k}{k-1} \right] }^q + 2 \phi _k^{(2)} {\left[ \frac{2}{k} + \frac{1}{2} \right] }^{q-2} \\\le & {} 2 {\left[ \frac{k}{k-1} \right] }^q + \left[ \frac{4 \ln {(k-2)}}{k} + \frac{\pi ^2 - 6}{3} \right] {\left[ \frac{2}{k} + \frac{1}{2} \right] }^{q-2} \;\; = \;\; \gamma _k. \end{aligned}$$

$\square$

Define the weights by

$$\begin{aligned} \omega _k^q := {\left\{ \begin{array}{ll} ~ 1, &{} \text {if} ~ k = 0 \\ |k|^q, &{} \text {if} ~ k \ne 0. \end{array}\right. } \end{aligned}$$

Lemma 6.2

(Quadratic estimates). Given a decay rate $q \ge 2$ and $M \ge 6$. For $k \in \mathbb {Z}$, define the quadratic asymptotic estimates $\alpha _k^{(2)} = \alpha _k^{(2)}(q,M)$ by

$$\begin{aligned} \alpha _k^{(2)} \,\mathop {=}\limits ^{{{\mathrm{def}}}}\,\left\{ \begin{array}{ll} \displaystyle 1 + 2 \sum _{k_1=1}^M \frac{1}{\omega _{k_1}^{2q}} + \frac{2}{M^{2q-1}(2q-1)}, &{} \text {for} ~ k=0 \\ \displaystyle \sum _{k_1 = 1}^{M} \frac{2 \omega _k^q}{\omega _{k_1}^q \omega _{k+k_1}^q} + \frac{2 \omega _k^q}{(k+M+1)^q M^{q-1} (q-1)} \\ \qquad \displaystyle +\, 2 + \sum _{k_1 = 1}^{k-1} \frac{\omega _k^q}{\omega _{k_1}^q \omega _{k-k_1}^q}, &{} \text {for} ~ 1 \le k \le M-1 \\ \displaystyle { 2 + 2 \sum _{k_1=1}^M \frac{1}{\omega _{k_1}^q} + \frac{2}{M^{q-1}(q-1)} + \gamma _M, } &{} \text {for} ~ k \ge M, \end{array} \right. \end{aligned}$$

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and for $k < 0$,

$$\begin{aligned} \alpha _k^{(2)} \,\mathop {=}\limits ^{{{\mathrm{def}}}}\,\alpha _{|k|}^{(2)}. \end{aligned}$$

Then, for any $k \in \mathbb {Z}$,

$$\begin{aligned} \sum _{\mathop {k_j \in \mathbb {Z}}\limits ^{k_1 + k_2 = k}} \frac{1}{\omega ^q_{k_1} \omega ^q_{k_2}} \le \frac{\alpha ^{(2)}_k}{\omega ^q_k}. \end{aligned}$$

Proof

For $k=0$,

$$\begin{aligned} \sum _{\mathop {k_j \in \mathbb {Z}}\limits ^{k_1 + k_2 = 0}} \frac{1}{\omega ^q_{k_1} \omega ^q_{k_2}}= & {} 1 + 2 \sum _{k_1 = 1}^{M} \frac{1}{\omega _{k_1}^{2q}} + 2 \sum _{k_1 = M+1}^{\infty } \frac{1}{\omega _{k_1}^{2q}} \le 1 + 2 \sum _{k_1 = 1}^{M} \frac{1}{\omega _{k_1}^{2q}} + \int _M^\infty \frac{\mathrm{d}x}{x^{2q}} \\\le & {} 1 + 2 \sum _{k_1 = 1}^{M} \frac{1}{\omega _{k_1}^{2q}} + \frac{2}{M^{2q-1} (2q - 1)} = \frac{\alpha _0^{(2)}}{\omega _0^q}. \end{aligned}$$

For $1 \le k \le M-1$, and recalling that the one-dimensional weights (2),

$$\begin{aligned}&\sum _{\mathop {k_j \in \mathbb {Z}}\limits ^{k_1 + k_2 = k}} \frac{1}{\omega ^q_{k_1} \omega ^q_{k_2}} \\&\quad = \frac{1}{\omega ^q_k} \left[ \sum _{k_1 = 1}^{M} \frac{2 \omega ^q_k}{\omega ^q_{k_1} \omega ^q_{k+k_1}} + \sum _{k_1 = M+1}^{\infty } \frac{2 \omega ^q_k}{\omega ^q_{k_1} \omega ^q_{k+k_1}} + \frac{2}{\omega ^q_0} + \sum _{k_1 = 1}^{k-1} \frac{\omega ^q_k}{\omega ^q_{k_1} \omega ^q_{k-k_1}} \right] \\&\quad \le \frac{1}{\omega ^q_k} \left[ \sum _{k_1 = 1}^{M} \frac{2 \omega _k^q}{\omega _{k_1}^q \omega _{k+k_1}^q} + \frac{2 \omega _k^q}{(k+M+1)^q} \int _M^\infty \frac{\mathrm{d}x}{x^q} + 2 + \sum _{k_1 = 1}^{k-1} \frac{\omega _k^q}{\omega _{k_1}^q \omega _{k-k_1}^q} \right] \\&\quad \le \frac{1}{\omega ^q_k} \left[ \sum _{k_1 = 1}^{M} \frac{2 \omega _k^q}{\omega _{k_1}^q \omega _{k+k_1}^q} + \frac{2 \omega _k^q}{(k+M+1)^q M^{q-1} (q-1)} + 2 + \sum _{k_1 = 1}^{k-1} \frac{\omega _k^q}{\omega _{k_1}^q \omega _{k-k_1}^q} \right] = \frac{\alpha _k^{(2)}}{\omega ^q_k}. \end{aligned}$$

Finally, for $k \ge M$, one gets from Lemma 6.1 that

$$\begin{aligned} \sum _{\mathop {k_j \in \mathbb {Z}}\limits ^{k_1 + k_2 = k}} \frac{1}{\omega ^q_{k_1} \omega ^q_{k_2}}= & {} \frac{1}{\omega ^q_k} \left[ 2 \sum _{k_1 = 1}^{\infty } \frac{\omega ^q_k}{\omega ^q_{k_1} \omega ^q_{k+k_1}} + \frac{2}{\omega ^q_0} + \sum _{k_1 = 1}^{k-1} \frac{\omega ^q_k}{\omega ^q_{k_1} \omega ^q_{k-k_1}} \right] \\\le & {} \frac{1}{\omega ^q_k} \left[ 2 \sum _{k_1=1}^M \frac{1}{\omega _{k_1}^q} + 2 \sum _{k_1=M+1}^\infty \frac{1}{\omega _{k_1}^q} + \frac{2}{\omega ^q_0} + \gamma _k \right] \\\le & {} \frac{1}{\omega ^q_k} \left[ 2 \sum _{k_1=1}^M \frac{1}{\omega _{k_1}^q} + 2 \int _{M}^\infty \frac{\mathrm{d}x}{x^q} + \frac{2}{\omega ^q_0} + \gamma _M \right] \\\le & {} \frac{1}{\omega ^q_k} \left[ \sum _{k_1=1}^M \frac{2}{\omega _{k_1}^q} + \frac{2}{M^{q-1}(q-1)} + 2 + \gamma _M \right] = \frac{\alpha _k^{(2)}}{\omega ^q_k}. \end{aligned}$$

$\square$

Lemma 6.3

For any $k \in \mathbb {Z}$ with $|k| \ge M \ge 6$, we have that $\alpha _k^{(2)} \le \alpha _M^{(2)}$.

Proof

For $k \ge 6$, the fact that $\frac{\ln (k-1)}{k+1} \le \frac{\ln (k-2)}{k}$ implies that $\gamma _{k+1}(q) \le \gamma _{k}(q)$. By definition of $\alpha _k^{(2)}$, for $|k| \ge M$, one gets that $\alpha _k^{(2)} \le \alpha _M^{(2)}$. $\square$

Lemma 6.4

(Cubic estimates). Given $q \ge 2$ and $M \ge 6$. Let

$$\begin{aligned} \Sigma _a^* := \sum _{k_1=1}^{M-1} \frac{\alpha _{k_1}^{(2)} M^q }{\omega _{k_1}^q \bigl ( M - k_1 \bigr )^{q} } + \alpha _M^{(2)} \left( \gamma _M - \sum _{k_1=1}^{M-1} \frac{1}{\omega _{k_1}^q} \right) , \end{aligned}$$

$\tilde{\alpha }_{M}^{(2)} := \max \left\{ \alpha _k^{(2)} \mid k = 0, \ldots , M \right\}$, $\Sigma _b^* := \tilde{\alpha }_{M}^{(2)} \gamma _M$ and $\Sigma ^* := \min \left\{ \Sigma _a^*, \Sigma _b^* \right\}$. Define the cubic asymptotic estimates $\alpha _k^{(3)} = \alpha _k^{(3)}(s,M)$ by

$$\begin{aligned} \alpha _k^{(3)} \! \,\mathop {=}\limits ^{{{\mathrm{def}}}}\,\! \left\{ \!\! \begin{array}{ll} \displaystyle { \alpha _0^{(2)} + 2 \sum _{k_1=1}^{M-1} \frac{\alpha _{k_1}^{(2)}}{\omega _{k_1}^{2q}} + \frac{2 \alpha _M^{(2)} }{(M-1)^{2q-1}(2q-1)}, } &{} \mathrm{for} ~ k=0 \\ \displaystyle { \sum _{k_1=1}^{M-k} \! \frac{\alpha _{k+k_1}^{(2)} \omega _k^q }{\omega _{k_1}^q \omega _{k+k_1}^q} + \frac{\alpha _M^{(2)} \omega _k^q }{(M+1)^q(M-k)^{q-1}(q-1)} + \sum _{k_1=1}^{k-1} \frac{\alpha _{k_1}^{(2)} \omega _k^q}{\omega _{k_1}^q \omega _{k-k_1}^q} }\\ \displaystyle \qquad + \sum _{k_1=1}^M \frac{\alpha _{k_1}^{(2)} \omega _k^q}{\omega _{k_1}^q \omega _{k+k_1}^q} + \frac{\alpha _M^{(2)} \omega _k^q }{(M+k+1)^qM^{q-1}(q-1)} + \alpha _{k}^{(2)}+ \alpha _0^{(2)},\\ \qquad \qquad &{} \mathrm{for} ~ 1 \le k \le M\!-\!1 \\ \displaystyle { \alpha _M^{(2)} \sum _{k_1=1}^M \frac{1}{\omega _{k_1}^q} + \frac{2 \alpha _M^{(2)}}{M^{q-1}(q-1)} + \Sigma ^* + \sum _{k_1=1}^M \frac{\alpha _{k_1}^{(2)}}{\omega _{k_1}^q} + \alpha _M^{(2)} + \alpha _0^{(2)}, } &{} \mathrm{for} ~ k \ge M \end{array} \right. \end{aligned}$$

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and for $k < 0$,

$$\begin{aligned} \alpha _k^{(3)} \,\mathop {=}\limits ^{{{\mathrm{def}}}}\,\alpha _{|k|}^{(3)}. \end{aligned}$$

Then, for any $k \in \mathbb {Z}$,

$$\begin{aligned} \sum _{\mathop {k_j \in \mathbb {Z}}\limits ^{k_1 + k_2 + k_3 = k}} \frac{1}{\omega ^q_{k_1} \omega ^q_{k_2} \omega ^q_{k_3}} \le \frac{\alpha ^{(3)}_k}{\omega ^q_k}. \end{aligned}$$

Moreover, $\alpha _k^{(3)} \le \alpha _M^{(3)}$, for all $k \ge M$.

Proof

In what follows, the estimates are obtained similarly as in the proof of Lemma 6.2 with the difference that we often use the fact $\alpha _k^{(2)} \le \alpha _M^{(2)}$, for all $k \ge M$ (see, e.g., Remark A.1 in [14]). For $k=0$,

$$\begin{aligned} \sum _{\mathop {k_j \in \mathbb {Z}}\limits ^{k_1 + k_2 + k_3 = 0}} \frac{1}{\omega ^q_{k_1} \omega ^q_{k_2} \omega ^q_{k_3}}\le & {} \alpha _0^{(2)} + 2 \sum _{k_1 = 1}^{M-1} \frac{\alpha _{k_1}^{(2)}}{\omega _{k_1}^{2q}} + \frac{2 \alpha _M^{(2)}}{(M - 1)^{2q-1} (2q-1)} = \frac{\alpha ^{(3)}_0}{\omega ^q_0}. \end{aligned}$$

For $k > 0$,

$$\begin{aligned} \sum _{\mathop {k_j \in \mathbb {Z}}\limits ^{k_1 + k_2 + k_3 = k}} \frac{1}{\omega ^q_{k_1}\omega ^q_{k_2} \omega ^q_{k_3}}\le & {} \sum _{k_1 = 1}^{\infty } \left[ \frac{1}{\omega ^q_{k_1}} \frac{\alpha _{k+k_1}^{(2)}}{\omega ^q_{k+k_1}} \right] + \sum _{k_1 = 1}^{k-1} \left[ \frac{1}{\omega ^q_{k_1}} \frac{\alpha _{k-k_1}^{(2)}}{\omega ^q_{k-k_1}} \right] + \sum _{k_1 = 1}^{\infty } \left[ \frac{1}{\omega ^q_{k+k_1}} \frac{\alpha _{k_1}^{(2)}}{\omega ^q_{k_1}} \right] \\&+\, \frac{1}{\omega ^q_0} \frac{\alpha _{k}^{(2)}}{\omega ^q_{k}} + \frac{1}{\omega ^q_k} \frac{\alpha _{0}^{(2)}}{\omega ^q_{0}}. \end{aligned}$$

Consider $k \in \{ 1, \ldots , M-1 \}$. Since $\alpha _k^{(2)} \le \alpha _M^{(2)}$, for all $k \ge M$ by Lemma 6.3, we have

$$\begin{aligned} \sum _{k_1=1}^{\infty } \frac{\alpha _{k+k_1}^{(2)} }{\omega _{k_1}^q \omega _{k+k_1}^q} \le \frac{1}{\omega _k^q} \left[ \sum _{k_1=1}^{M-k} \! \frac{\alpha _{k+k_1}^{(2)} \omega _k^q }{\omega _{k_1}^q \omega _{k+k_1}^q} + \frac{\alpha _M^{(2)} \omega _k^q }{(M+1)^q(M-k)^{q-1}(q-1)} \right] \! . \end{aligned}$$

Similarly,

$$\begin{aligned} \sum _{k_1=1}^{\infty } \frac{\alpha _{k_1}^{(2)}}{\omega _{k_1}^q \omega _{k+k_1}^q} \le \frac{1}{\omega _k^q} \left[ \sum _{k_1=1}^M \frac{\alpha _{k_1}^{(2)} \omega _k^q}{\omega _{k_1}^q \omega _{k+k_1}^q} + \frac{\alpha _M^{(2)} \omega _k^q }{(M+k+1)^qM^{q-1}(q-1)} \right] . \end{aligned}$$

From the definition of $\alpha _k^{(3)}$ for $k \in \{1, \ldots ,M-1\}$, one gets that

$$\begin{aligned} \sum _{\mathop {k_j \in \mathbb {Z}}\limits ^{k_1 + k_2 + k_3 = k}} \frac{1}{\omega ^q_{k_1}\omega ^q_{k_2} \omega ^q_{k_3}} \le \frac{ \alpha _k^{(3)}}{\omega _k^q}. \end{aligned}$$

For $k \ge M$, using again that $\alpha _k^{(2)} \le \alpha _M^{(2)}$ by Lemma 6.3, one gets that

$$\begin{aligned} \sum _{k_1=1}^{\infty } \frac{\alpha _{k+k_1}^{(2)} }{\omega _{k_1}^q \omega _{k+k_1}^q} \le \frac{1}{\omega _k^q} \left[ \alpha _M^{(2)} \sum _{k_1=1}^M \frac{1}{\omega _{k_1}^q} + \frac{\alpha _M^{(2)}}{M^{q-1}(q-1)} \right] . \end{aligned}$$

Using Lemma 6.1,

$$\begin{aligned} \sum _{k_1=1}^{k-1} \frac{\alpha _{k_1}^{(2)}}{\omega _{k_1}^q \omega _{k-k_1}^q}= & {} \sum _{k_1=1}^{M-1} \frac{\alpha _{k_1}^{(2)}}{\omega _{k_1}^q \omega _{k-k_1}^q} + \frac{1}{\omega _k^q} \sum _{k_1=M}^{k-1} \frac{\omega _k^q \alpha _{k_1}^{(2)}}{\omega _{k_1}^q \omega _{k-k_1}^q} \\\le & {} \frac{1}{\omega _k^q} \sum _{k_1=1}^{M-1} \frac{\alpha _{k_1}^{(2)}}{\omega _{k_1}^q \bigl (1-\frac{k_1}{k}\bigr )^{q}} + \frac{\alpha _M^{(2)}}{\omega _k^q} \sum _{k_1=M}^{k-1} \frac{\omega _k^q}{\omega _{k_1}^q \omega _{k-k_1}^q} \\\le & {} \frac{1}{\omega _k^q} \left[ \sum _{k_1=1}^{M-1} \frac{\alpha _{k_1}^{(2)}}{\omega _{k_1}^q \bigl ( 1 \!-\! \frac{k_1}{M} \bigr )^{q} } \!+\! \alpha _M^{(2)} \left( \sum _{k_1=1}^{k-1} \frac{\omega _k^q}{\omega _{k_1}^q \omega _{k-k_1}^q} \!-\! \sum _{k_1=1}^{M-1} \frac{\omega _k^q}{\omega _{k_1}^q \omega _{k-k_1}^q} \right) \right] \\\le & {} \frac{1}{\omega _k^q} \left[ \sum _{k_1=1}^{M-1} \frac{\alpha _{k_1}^{(2)} M^q }{\omega _{k_1}^q \bigl ( M - k_1 \bigr )^{q} } + \alpha _M^{(2)} \left( \gamma _M - \sum _{k_1=1}^{M-1} \frac{1}{\omega _{k_1}^q} \right) \right] = \frac{1}{\omega _k^q} \Sigma _a^* . \end{aligned}$$

Hence,

$$\begin{aligned} \sum _{k_1=1}^{k-1} \frac{\alpha _{k_1}^{(2)}}{\omega _{k_1}^q \omega _{k-k_1}^q} \le \frac{\tilde{\alpha }_{M}^{(2)} }{\omega _k^q} \gamma _M = \frac{1}{\omega _k^q} \Sigma _b^*. \end{aligned}$$

Recalling that $\Sigma ^* = \min \left\{ \Sigma _a^*, \Sigma _b^* \right\}$, one gets that $\displaystyle \sum \nolimits _{k_1=1}^{k-1} \frac{\alpha _{k_1}^{(2)}}{\omega _{k_1}^q \omega _{k-k_1}^q} \le \frac{1}{\omega _k^q} \Sigma ^*$. Also,

$$\begin{aligned} \sum _{k_1=1}^{\infty } \frac{\alpha _{k_1}^{(2)}}{\omega _{k_1}^q \omega _{k+k_1}^q} \le \frac{1}{\omega _k^q} \left[ \sum _{k_1=1}^M \frac{\alpha _{k_1}^{(2)}}{\omega _{k_1}^q} + \frac{\alpha _M^{(2)}}{M^{q-1} (q-1)} \right] . \end{aligned}$$

Combining the above inequalities, we get, for the case $k \ge M$,

$$\begin{aligned}&\sum _{\mathop {k_j \in \mathbb {Z}}\limits ^{k_1 + k_2 + k_3 = k}} \frac{1}{\omega ^q_{k_1}\omega ^q_{k_2} \omega ^q_{k_3}} \\&\quad \quad \le \frac{1}{\omega _k^q} \left[ \alpha _M^{(2)} \sum _{k_1=1}^M \frac{1}{\omega _{k_1}^q} + \frac{2 \alpha _M^{(2)}}{M^{q-1}(q-1)} + \Sigma ^* + \sum _{k_1=1}^M \frac{\alpha _{k_1}^{(2)}}{\omega _{k_1}^q} \!+\! \alpha _M^{(2)} + \alpha _0^{(2)} \right] \!=\! \dfrac{\alpha _k^{(3)}}{\omega _k^q}. \end{aligned}$$

$\square$

Lemma 6.5

For any $k \in \mathbb {Z}$ with $|k| \ge M \ge 6$, we have that $\alpha _k^{(3)} \le \alpha _M^{(3)}$.

Proof

For $k \ge 6$, the fact that $\frac{\ln (k-1)}{k+1} \le \frac{\ln (k-2)}{k}$ implies that $\gamma _{k+1}(q) \le \gamma _{k}(q)$. By definition of $\alpha _k^{(3)}$, for $|k| \ge M$, one gets that $\alpha _k^{(3)} \le \alpha _M^{(3)}$. $\square$

Lemma 6.6

Given $q \ge 2$ and $6 \le {\bar{M}}\le M$, define for $0 \le k \le {\bar{M}}-1$

$$\begin{aligned} \varepsilon _{k}^{(3)} = \varepsilon _{k}^{(3)}(q, {\bar{M}}, M):= & {} \sum _{k_1={\bar{M}}}^{M-k} \frac{\alpha _{k+k_1}^{(2)}}{ \omega ^q_{k_1} \omega ^q_{k+k_1}} + \sum _{k_1={\bar{M}}}^{M+k} \frac{\alpha _{k_1-k}^{(2)}}{\omega ^q_{k_1} \omega ^q_{k_1-k}}\nonumber \\&+\, \frac{ \alpha _{M}^{(2)}}{(M+1)^q(q-1)} \left[ \frac{1}{(M-k)^{q-1}} + \frac{1}{(M+k)^{q-1}} \right] \nonumber \\ \end{aligned}$$

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and for $k < 0$

$$\begin{aligned} \varepsilon _{k}^{(3)}(q,{\bar{M}},M) := \varepsilon _{|k|}^{(3)}(q,{\bar{M}},M). \end{aligned}$$

Fix $0 \le |k| \le {\bar{M}}-1$ and $\ell \in \{1,2,3\}$. Then, we have that

$$\begin{aligned} \sum _{\mathop {\max \{ |k_1|, \ldots , |k_{\ell }| \} \ge {\bar{M}}}\limits ^{k_1+k_2 +k_3=k}} \frac{1}{\omega ^q_{k_1}\omega ^q_{k_2} \omega ^q_{k_3}} \le \ell \varepsilon _k^{(3)}. \end{aligned}$$

Proof

We have that

$$\begin{aligned} \sum _{\mathop {\max \{ |k_1|, \ldots , |k_{\ell }| \} \ge {\bar{M}}}\limits ^{k_1+k_2 +k_3=k}} \frac{1}{\omega ^q_{k_1}\omega ^q_{k_2} \omega ^q_{k_3}} \le \ell \sum _{\mathop { |k_1| \ge {\bar{M}}}\limits ^{k_1+k_2 +k_3=k}} \frac{1}{\omega ^q_{k_1}\omega ^q_{k_2} \omega ^q_{k_3}}, \end{aligned}$$

and

$$\begin{aligned} \sum _{\mathop { |k_1| \ge {\bar{M}}}\limits ^{k_1+k_2 +k_3=k}} \frac{1}{\omega ^q_{k_1}\omega ^q_{k_2} \omega ^q_{k_3}}= & {} \sum _{k_1=-\infty }^{-{\bar{M}}} \frac{1}{\omega ^q_{k_1}} \sum _{k_2 + k_3=k-k_1} \frac{1}{\omega ^q_{k_2}\omega ^q_{k_2} \omega ^q_{k_3}} \\&+ \sum _{k_1={\bar{M}}}^{\infty } \frac{1}{\omega ^q_{k_1}} \sum _{k_2 + k_3 = k-k_1} \frac{1}{\omega ^q_{k_2}\omega ^q_{k_2} \omega ^q_{k_3}} \\\le & {} \sum _{k_1={\bar{M}}}^{\infty } \left[ \frac{\alpha _{k+k_1}^{(2)}}{ \omega ^q_{k_1} \omega ^q_{k+k_1}} + \frac{\alpha _{k_1-k}^{(2)}}{\omega ^q_{k_1} \omega ^q_{k_1-k}} \right] \\\le & {} \sum _{k_1={\bar{M}}}^{M-k} \frac{\alpha _{k+k_1}^{(2)}}{ \omega ^q_{k_1} \omega ^q_{k+k_1}} + \alpha _{M}^{(2)} \sum _{k_1=M-k+1}^{\infty } \frac{1}{ \omega ^q_{k_1} \omega ^q_{k+k_1}} \\&+ \sum _{k_1={\bar{M}}}^{M+k} \frac{\alpha _{k_1-k}^{(2)}}{\omega ^q_{k_1} \omega ^q_{k_1-k}} + \alpha _{M}^{(2)} \sum _{k_1=M+k+1}^{\infty } \frac{1}{\omega ^q_{k_1} \omega ^q_{k_1-k}} \\\le & {} \sum _{k_1={\bar{M}}}^{M-k} \frac{\alpha _{k+k_1}^{(2)}}{ \omega ^q_{k_1} \omega ^q_{k+k_1}} + \sum _{k_1={\bar{M}}}^{M+k} \frac{\alpha _{k_1-k}^{(2)}}{\omega ^q_{k_1} \omega ^q_{k_1-k}}\\&+ \frac{ \alpha _{M}^{(2)}}{(M+1)^q(q-1)} \left[ \frac{1}{(M-k)^{q-1}} + \frac{1}{(M+k)^{q-1}} \right] =\varepsilon _k^{(3)}. \end{aligned}$$

$\square$

Appendix 2: Coefficients Used to Define the Bounds Y and Z

The coefficients are presented in Tables 3, 4, 5 and 6