Asymptotic analysis of the conventional and invariant schemes for the method of fundamental solutions applied to potential problems in doubly-connected regions

Abstract

The aim of this paper is to develop mathematical theory of the conventional and invariant schemes for the method of fundamental solutions used to solve potential problems in doubly-connected regions. Particularly, we prove that an approximate solution actually exists uniquely under some conditions, and that the error decays exponentially when the boundary data are analytic, and algebraically when they are not analytic but belong to some Sobolev spaces. Moreover, we present results of several numerical experiments in order to show the sharpness of our error estimate.

Appendices

A Proof of Lemma 2

First, we bound the norm $\Vert q - q^{(N)}\Vert _{\mathbb {Y}_{\epsilon , s}}^2$ as follows:

$$\begin{aligned} \Vert q - q^{(N)}\Vert _{\mathbb {Y}_{\epsilon , s}}^2&\le \sum _{\nu =1}^{2}[ T_1^{(\nu )} + (2\pi )^{s-1} ( T_2^{(\nu )} + 2T_3^{(\nu )} + 2T_4^{(\nu )} ) ], \end{aligned}$$

where

$$\begin{aligned} T_1^{(\nu )}&= |\hat{q}_\nu (0) - \hat{q}_\nu ^{(N)}(0)|^2, \\ T_2^{(\nu )}&= \sum _{n \in \varLambda _N' {\setminus } \{0\}} |\hat{q}_\nu (n) - \hat{q}_\nu ^{(N)}(n)|^2 (\epsilon \xi (\nu , \nu ))^{2|n|} |n|^{2(s-1)},\\ T_3^{(\nu )}&= \sum _{n \in \mathbb {Z}{\setminus } \varLambda _N'} |\hat{q}_\nu ^{(N)}(n)|^2 (\epsilon \xi (\nu , \nu ))^{2|n|} |n|^{2(s-1)},\\ T_4^{(\nu )}&= \sum _{n \in \mathbb {Z}{\setminus } \varLambda _N'} |\hat{q}_\nu (n)|^2 (\epsilon \xi (\nu , \nu ))^{2|n|} |n|^{2(s-1)} \end{aligned}$$

for $\nu = 1, 2$. Here, we set $\varLambda _N' = \{p \in \mathbb {Z}\mid -N/2 < p \le N/2\}$. In the following, we provide estimates for $T_j^{(1)}$ for $j = 1, 2, 3, 4$.

Utilizing the relations (3.7), the Fourier coefficients $\hat{q}_\nu ^{(N)}(n)$ can be obtained explicitly as follows:

$$\begin{aligned} \begin{pmatrix} \hat{q}_1^{(N)}(n) \\[1ex] \hat{q}_2^{(N)}(n) \end{pmatrix} = \frac{1}{\det \varPhi ^\mathrm {C}(n)} \sum _{m \equiv n} \sum _{l \equiv n} \begin{pmatrix} \varUpsilon _1(m, l) \hat{q}_1(m) + \varUpsilon _2(m, l) \hat{q}_2(m) \\[1ex] \varUpsilon _3(m, l) \hat{q}_1(m) + \varUpsilon _4(m, l) \hat{q}_2(m) \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned} \varUpsilon _1(m, l)&= \hat{G}_{11}(m) \hat{G}_{22}(l) - \hat{G}_{21}(m) \hat{G}_{12}(l), \\ \varUpsilon _2(m, l)&= \hat{G}_{12}(m) \hat{G}_{22}(l) - \hat{G}_{22}(m) \hat{G}_{12}(l), \\ \varUpsilon _3(m, l)&= -\hat{G}_{11}(m) \hat{G}_{21}(l) + \hat{G}_{21}(m) \hat{G}_{11}(l), \\ \varUpsilon _4(m, l)&= -\hat{G}_{12}(m) \hat{G}_{21}(l) + \hat{G}_{22}(m) \hat{G}_{11}(l). \end{aligned}$$

We will employ the following proposition without proof.

Proposition 8

• (i) There exist some positive constants $C_j$ ($j = 1, 2, 3, 4$) such that

  $$\begin{aligned}&|\varUpsilon _1(m, l)| \le \frac{C_1}{\underline{m} \cdot \underline{l}} \left( \frac{\rho _1}{R_1}\right) ^{|m|} \left( \frac{R_2}{\rho _2}\right) ^{|l|}, \quad |\varUpsilon _2(m, l)| \le \frac{C_2}{\underline{m} \cdot \underline{l}} \left( \frac{R_2}{\rho _2}\right) ^{|m|+|l|}, \\&|\varUpsilon _3(m, l)| \le \frac{C_3}{\underline{m} \cdot \underline{l}} \left( \frac{\rho _1}{R_1}\right) ^{|m|+|l|}, \quad |\varUpsilon _4(m, l)| \le \frac{C_4}{\underline{m} \cdot \underline{l}} \left( \frac{R_2}{\rho _2}\right) ^{|m|} \left( \frac{\rho _1}{R_1}\right) ^{|l|}. \end{aligned}$$

• (ii) There exists some positive constant $C_{\rho _1,\,\rho _2,\,R_1,\,R_2}$ such that

  $$\begin{aligned} \frac{1}{(\det \varPhi ^\mathrm {C}(0))^2} \le C_{\rho _1,\,\rho _2,\,R_1,\,R_2} \end{aligned}$$

  holds for all $N \in \mathbb {N}$.

• (iii) There exists some positive constant $C_{\rho _1, \rho _2, R_1, R_2}$ such that

  $$\begin{aligned} \frac{1}{(\det \varPhi ^\mathrm {C}(n))^2} \le C_{\rho _1, \rho _2, R_1, R_2} |n|^4 \left( \frac{R_1}{\rho _1}\right) ^{2|n|} \left( \frac{\rho _2}{R_2}\right) ^{2|n|} \end{aligned}$$

  holds for all $N \in \mathbb {N}$, and all $n \in \varLambda _N' {\setminus } \{0\}$.

• (iv) For all $(\epsilon , s) \in \ ]0, +\infty [ \times \mathbb {R}$ with $(\epsilon , s) < (1, -1)$, there exists some positive constant $C_{\epsilon , s}$ such that

  $$\begin{aligned} \sum _{m \in I(p)} |m|^s \epsilon ^{|m|} \le C_{\epsilon , s} N^s \epsilon ^{N-|p|} \end{aligned}$$

  holds for all $N \in \mathbb {N}$, and all $p \in \varLambda _N'$.

• (v) For all $(\epsilon , s) \in \ ]0, 1[ \times \mathbb {R}$, there exists some positive constant $C_{\epsilon , s}$ such that

  $$\begin{aligned} \max _{p \in \varLambda _N' {\setminus } \{0\}} \left( \left( \frac{N}{|p|}\right) ^s \epsilon ^{N-2|p|} \right) \le C_{\epsilon , s} \end{aligned}$$

  holds for all $N \in \mathbb {N}$.

$T_1^{(1)}$ can be bounded as

$$\begin{aligned} T_1^{(1)} \le T_{11}^{(1)} + T_{12}^{(1)} + T_{13}^{(1)}, \end{aligned}$$

where

$$\begin{aligned} T_{11}^{(1)}&= \frac{3}{(\det \varPhi ^\mathrm {C}(0))^2} \left( \sum _{m \in I(0)} \sum _{l \equiv 0} |\varUpsilon _1(m, l)| |\hat{q}_1(0)| \right) ^2,\\ T_{12}^{(1)}&= \frac{3}{(\det \varPhi ^\mathrm {C}(0))^2} \left( \sum _{m \in I(0)} \sum _{l \equiv 0} |\varUpsilon _1(m, l)| |\hat{q}_1(m)| \right) ^2, \\ T_{13}^{(1)}&= \frac{3}{(\det \varPhi ^\mathrm {C}(0))^2} \left( \sum _{\begin{array}{c} m, l \equiv 0 \\ m \ne l \end{array}} |\varUpsilon _2(m, l)| |\hat{q}_2(m)| \right) ^2. \end{aligned}$$

By using Proposition 8 (i), (ii), and (iv), we have that

$$\begin{aligned} T_{11}^{(1)}&\le C_{11}^{(1)} \left( \sum _{m \in I(0)} \sum _{l \equiv 0} \frac{1}{\underline{m} \cdot \underline{l}} \left( \frac{\rho _1}{R_1}\right) ^{|m|} \left( \frac{R_2}{\rho _2}\right) ^{|l|} \right) ^2 |\hat{q}_1(0)|^2 \\&\le C_{11}^{(1)} N^{-2} \left( \frac{\rho _1}{R_1}\right) ^{2N} \Vert q_1\Vert _{\delta \rho _1/R_1, t-1}^2. \end{aligned}$$

By assumption, we have that $\delta r^2 \le \epsilon$. Therefore, the following inequalities can be obtained:

By using Proposition 8 (i), (ii), and (iv), we have that

$$\begin{aligned} T_{12}^{(1)}&\le C_{12}^{(1)} \left( \sum _{m \in I(0)} \sum _{l \equiv 0} \frac{1}{\underline{m} \cdot \underline{l}} \left( \frac{\rho _1}{R_1}\right) ^{|m|} \left( \frac{R_2}{\rho _2}\right) ^{|l|} |\hat{q}_1(m)| \right) ^2 \\&\le C_{12}^{(1)} \sum _{m \in I(0)} \frac{1}{\delta ^{2|m|}} \frac{1}{\underline{m}^{2t}} \sum _{m \in I(0)} |\hat{q}_1(m)|^2 \left( \frac{\delta \rho _1}{R_1}\right) ^{2|m|} \underline{m}^{2(t-1)}\\&\le C_{12}^{(1)} N^{-2t} \frac{1}{\delta ^{2N}} \Vert q_1\Vert _{\delta \rho _1/R_1, t-1}^2. \end{aligned}$$

The assumption that $1/\delta \le \epsilon$ implies that

Then, $T_{13}^{(1)}$ is bounded by splitting it into the following three terms:

$$\begin{aligned} T_{13}^{(1)} \le T_{131}^{(1)} + T_{132}^{(1)} + T_{133}^{(1)}, \end{aligned}$$

where

$$\begin{aligned} T_{131}&= \frac{9}{(\det \varPhi ^\mathrm {C}(0))^2} \left( \sum _{l \in I(0)} |\varUpsilon _2(0, l)| |\hat{q}_2(0)| \right) ^2, \\ T_{132}&= \frac{9}{(\det \varPhi ^\mathrm {C}(0))^2} \left( \sum _{m \in I(0)} |\varUpsilon _2(m, 0)| |\hat{q}_2(m)| \right) ^2, \\ T_{133}&= \frac{9}{(\det \varPhi ^\mathrm {C}(0))^2} \left( \sum _{m, l \in I(0)} |\varUpsilon _2(m, l)| |\hat{q}_2(m)| \right) ^2. \end{aligned}$$

In the same manner as in deriving an estimate for $T_{11}^{(1)}$, $T_{131}^{(1)}$ can be bounded by Proposition 8 (i), (ii), and (iv) as follows:

In a similar manner as for estimating $T_{12}^{(1)}$, a bound for $T_{132}^{(1)}$ can be given by

by Proposition 8 (i), (ii), and (iv). Furthermore, $T_{133}^{(1)}$ can be estimated using Proposition 8 (i), (ii), and (iv) and the relation $\epsilon /\delta > (r/\delta )^2$, as follows:

$$\begin{aligned} T_{133}^{(1)}&\le C_{133}^{(1)} \left( \sum _{m \in I(0)} \sum _{l \in I(0)} \frac{1}{\underline{m} \cdot \underline{l}} \left( \frac{R_2}{\rho _2}\right) ^{|m| + |l|} |\hat{q}_2(m)| \right) ^2\\&\le C_{133}^{(1)} N^{2(s-t)} \left( \frac{\epsilon }{\delta }\right) ^N \Vert q_2\Vert _{\delta R_2/\rho _2, t-1}^2. \end{aligned}$$

Summarizing the above, we can obtain the following bound for $T_1^{(1)}$:

Next, we estimate $T_2^{(1)}$. Because we can estimate $|\hat{q}_1(n) - \hat{q}_1^{(N)}(n)|^2$ as

$$\begin{aligned} |\hat{q}_1(n) - \hat{q}_1^{(N)}(n)|^2&\le \frac{3}{(\det \varPhi ^\mathrm {C}(n))^2} \left[ \left( \sum _{m \in I(n)} \sum _{l \equiv n} |\varUpsilon _1(m, l)| |\hat{q}_1(n)| \right) ^2 \right. \\&\quad \left. + \left( \sum _{m \in I(n)} \sum _{l \equiv n} |\varUpsilon _1(m, l)| |\hat{q}_1(m)| \right) ^2\right. \\&\quad \left. + \left( \sum _{\begin{array}{c} m, l \equiv n \\ m \ne l \end{array}} |\varUpsilon _2(m, l)| |\hat{q}_2(m)| \right) ^2 \right] , \end{aligned}$$

we have that

$$\begin{aligned} T_2^{(1)}&\le T_{21}^{(1)} + T_{22}^{(1)} + T_{23}^{(1)}, \end{aligned}$$

where

$$\begin{aligned} T_{21}^{(1)}&= \sum _{n \in \varLambda _N' {\setminus } \{0\}} \frac{3}{(\det \varPhi ^\mathrm {C}(n))^2} \left( \sum _{m \in I(n)} \sum _{l \equiv n} |\varUpsilon _1(m, l)| |\hat{q}_1(n)| \right) ^2 \left( \frac{\epsilon \rho _1}{R_1}\right) ^{2|n|} |n|^{2(s-1)}, \\ T_{22}^{(1)}&= \sum _{n \in \varLambda _N' {\setminus } \{0\}} \frac{3}{(\det \varPhi ^\mathrm {C}(n))^2} \left( \sum _{m \in I(n)} \sum _{l \equiv n} |\varUpsilon _1(m, l)| |\hat{q}_1(m)| \right) ^2 \left( \frac{\epsilon \rho _1}{R_1}\right) ^{2|n|} |n|^{2(s-1)}, \\ T_{23}^{(1)}&= \sum _{n \in \varLambda _N' {\setminus } \{0\}} \frac{3}{(\det \varPhi ^\mathrm {C}(n))^2} \left( \sum _{\begin{array}{c} m, l \equiv n \\ m \ne l \end{array}} |\varUpsilon _2(m, l)| |\hat{q}_2(m)| \right) ^2 \left( \frac{\epsilon \rho _1}{R_1}\right) ^{2|n|} |n|^{2(s-1)}. \end{aligned}$$

Using Proposition 8 (i), (iii), and (iv), for $n \in \varLambda _N' {\setminus } \{0\}$, we have that

$$\begin{aligned}&\frac{1}{(\det \varPhi ^\mathrm {C}(n))^2} \left( \sum _{m \in I(n)} \sum _{l \equiv n} |\varUpsilon _1(m, l)| |\hat{q}_1(n)| \right) ^2 \\&\quad \le C_{21}^{(1)} |n|^4 \left( \frac{R_1}{\rho _1}\right) ^{2|n|} \left( \frac{\rho _2}{R_2}\right) ^{2|n|} \left( \sum _{m \in I(n)} \sum _{l \equiv n} \frac{1}{\underline{m} \cdot \underline{l}} \left( \frac{\rho _1}{R_1}\right) ^{|m|} \left( \frac{R_2}{\rho _2}\right) ^{|l|} \right) ^2 |\hat{q}_1(n)|^2 \\&\quad \le C_{21}^{(1)} |n|^2N^{-2}\left( \frac{\rho _1}{R_1}\right) ^{2(N-2|n|)} |\hat{q}_1(n)|^2, \end{aligned}$$

which yields that

$$\begin{aligned} T_{21}^{(1)}&\le C_{21}^{(1)} \sum _{n \in \varLambda _N' {\setminus } \{0\}} |n|^2 N^{-2} \left( \frac{\rho _1}{R_1}\right) ^{2(N-2|n|)} |\hat{q}_1(n)|^2 \left( \frac{\epsilon \rho _1}{R_1}\right) ^{2|n|} |n|^{2(s-1)} \\&\le C_{21}^{(1)} N^{2[-1+\max \{s-t+1, 0\}]} \left( \frac{\epsilon }{\delta }\right) ^N \Vert q_1\Vert _{\delta \rho _1/R_1, t-1}^2 A_{21}^{(1)}, \end{aligned}$$

where

$$\begin{aligned} A_{21}^{(1)} = \sup _{n \in \varLambda _N' {\setminus } \{0\}} \left[ \left( \frac{\delta r^2}{\epsilon }\right) ^{N-2|n|} |n|^{2(s-t+1)} N^{-2\max \{s-t+1, 0\}} \right] . \end{aligned}$$

Because this constant can be evaluated using Proposition 8 (v) as

we obtain that

By Proposition 8 (i), (iii), and (iv), we have that

$$\begin{aligned}&\frac{1}{(\det \varPhi ^\mathrm {C}(n))^2} \left( \sum _{m \in I(n)} \sum _{l \equiv n} |\varUpsilon _1(m, l)| |\hat{q}_1(m)| \right) ^2 \\&\quad \le C_{22}^{(1)} |n|^4 \left( \frac{R_1}{\rho _1}\right) ^{2|n|} \left( \frac{\rho _2}{R_2}\right) ^{2|n|} \left( \sum _{m \in I(n)} \sum _{l \equiv n} \frac{1}{\underline{m} \cdot \underline{l}} \left( \frac{\rho _1}{R_1}\right) ^{|m|} \left( \frac{R_2}{\rho _2}\right) ^{|l|} |\hat{q}_1(m)| \right) ^2 \\&\quad \le C_{22}^{(1)} |n|^2 N^{-2t} \frac{1}{\delta ^{2(N-|n|)}} \left( \frac{R_1}{\rho _1}\right) ^{2|n|} \sum _{m \in I(n)} |\hat{q}_1(m)|^2 \left( \frac{\delta \rho _1}{R_1}\right) ^{2|m|} \underline{m}^{2(t-1)} \end{aligned}$$

for $n \in \varLambda _N' {\setminus } \{0\}$. Therefore, $T_{22}^{(1)}$ can be bounded as follows:

$$\begin{aligned} T_{22}^{(1)}&\le C_{22}^{(1)} \sum _{n \in \varLambda _N' {\setminus } \{0\}} |n|^2 N^{-2t} \frac{1}{\delta ^{2(N-|n|)}} \left( \frac{R_1}{\rho _1}\right) ^{2|n|} \left( \frac{\epsilon \rho _1}{R_1}\right) ^{2|n|} |n|^{2(s-1)}\\&\quad \times \sum _{m \in I(n)} |\hat{q}_1(m)|^2 \left( \frac{\delta \rho _1}{R_1}\right) ^{2|m|} \underline{m}^{2(t-1)} \\&\le C_{22}^{(1)} N^{2[-t+\max \{s, 0\}]} \left( \frac{\epsilon }{\delta }\right) ^N \Vert q_1\Vert _{\delta \rho _1/R_1, t-1}^2 A_{22}^{(1)}, \end{aligned}$$

where

$$\begin{aligned} A_{22}^{(1)} = \sup _{n \in \varLambda _N' {\setminus } \{0\}} \left[ \left( \frac{1}{\epsilon \delta }\right) ^{N-2|n|} |n|^{2s} N^{-2\max \{s, 0\}}\right] , \end{aligned}$$

which can be bounded as

by Proposition 8 (v). Then, we obtain the following estimate for $T_{22}^{(1)}$:

Concerning the estimation of $T_{23}^{(1)}$, we use Proposition 8 (i), (iii), and (iv) to yield that

$$\begin{aligned}&\frac{1}{(\det \varPhi ^\mathrm {C}(n))^2} \left( \sum _{\begin{array}{c} m, l \equiv n \\ m \ne l \end{array}} |\varUpsilon _2(m, l)| |\hat{q}_2(m)| \right) ^2 \\&\quad \le C |n|^4 \left( \frac{R_1}{\rho _1}\right) ^{2|n|} \left( \frac{\rho _2}{R_2}\right) ^{2|n|} \left( \sum _{\begin{array}{c} m, l \equiv n \\ m \ne l \end{array}} \frac{1}{\underline{m} \cdot \underline{l}} \left( \frac{R_2}{\rho _2}\right) ^{|m|+|l|} |\hat{q}_2(m)|\right) ^2 \\&\quad \le C_{231}^{(1)} |n|^2 N^{-2} \left( \frac{R_1}{\rho _1}\right) ^{2|n|} \left( \frac{R_2}{\rho _2}\right) ^{2(N-|n|)} |\hat{q}_2(n)|^2 \\&\qquad + C_{232}^{(1)} |n|^2 \left( \frac{R_1}{\rho _1}\right) ^{2|n|} \frac{1}{\delta ^{2(N-|n|)}} N^{-2t} \sum _{m \in I(n)} |\hat{q}_2(m)|^2 \left( \frac{\delta R_2}{\rho _2}\right) ^{2|m|} \underline{m}^{2(t-1)} \\&\qquad + C_{233}^{(1)} |n|^4 N^{-2(t+1)} \left( \frac{R_1}{\rho _1}\right) ^{2|n|} \left( \frac{R_2}{\rho _2}\right) ^{2(N-2|n|)} \frac{1}{\delta ^{2(N-|n|)}}\\&\qquad \times \sum _{m \in I(n)} |\hat{q}_2(m)|^2 \left( \frac{\delta R_2}{\rho _2}\right) ^{2|m|} \underline{m}^{2(t-1)}, \end{aligned}$$

which implies that

$$\begin{aligned} T_{23}^{(1)}&\le T_{231}^{(1)} + T_{232}^{(1)} + T_{233}^{(1)}, \end{aligned}$$

where

$$\begin{aligned} T_{231}^{(1)}&= C_{231}^{(1)} \sum _{n \in \varLambda _N' {\setminus } \{0\}} \epsilon ^{2|n|} |n|^{2s} N^{-2} \left( \frac{R_2}{\rho _2}\right) ^{2(N-|n|)} |\hat{q}_2(n)|^2, \\ T_{232}^{(2)}&= C_{232}^{(1)} \sum _{n \in \varLambda _N' {\setminus } \{0\}} \epsilon ^{2|n|} |n|^{2s} \frac{1}{\delta ^{2(N-2|n|)}} N^{-2t} \sum _{m \in I(n)} |\hat{q}_2(m)|^2 \left( \frac{\delta R_2}{\rho _2}\right) ^{2|m|} \underline{m}^{2(t-1)}, \\ T_{233}^{(1)}&= C_{233}^{(1)} \sum _{n \in \varLambda _N' {\setminus } \{0\}} \epsilon ^{2|n|} |n|^{2(s+1)} N^{-2(t+1)} \left( \frac{R_2}{\rho _2}\right) ^{2(N-2|n|)} \frac{1}{\delta ^{2(N-|n|)}}\\&\quad \times \sum _{m \in I(n)} |\hat{q}_2(m)|^2 \left( \frac{\delta R_2}{\rho _2}\right) ^{2|m|} \underline{m}^{2(t-1)}. \end{aligned}$$

We estimate each of these quantities below. Concerning $T_{231}^{(1)}$, we have that

$$\begin{aligned} T_{231}^{(1)}&\le C_{231}^{(1)} N^{2[-1+\max \{s-t+1, 0\}]} \left( \frac{\epsilon }{\delta }\right) ^N \Vert q_2\Vert _{\delta R_2/\rho _2, t-1}^2 A_{231}^{(1)}, \end{aligned}$$

where

$$\begin{aligned} A_{231}^{(1)} = \sup _{n \in \varLambda _N' {\setminus } \{0\}} \left[ |n|^{2(s-t+1)} N^{-2\max \{s-t+1, 0\}} \left( \frac{\delta r^2}{\epsilon }\right) ^{N-2|n|} \right] . \end{aligned}$$

This can be bounded by virtue of Proposition 8 (v) as follows:

Therefore, we can determine an estimate for $T_{231}^{(1)}$ as

Concerning $T_{232}^{(1)}$, we have that

$$\begin{aligned} T_{232}^{(1)}&\le C_{232}^{(1)} N^{2[-t+\max \{s, 0\}]} \left( \frac{\epsilon }{\delta }\right) ^N \Vert q_2\Vert _{\delta R_2/\rho _2, t-1}^2 A_{232}^{(1)}, \end{aligned}$$

where

$$\begin{aligned} A_{232}^{(1)} = \sup _{n \in \varLambda _N' {\setminus } \{0\}} \left[ |n|^{2s} N^{-2\max \{s, 0\}} \left( \frac{1}{\delta \epsilon }\right) ^{N-2|n|} \right] . \end{aligned}$$

The constant $A_{232}^{(1)}$ can be bounded by using Proposition 8 (v), as follows:

Then, we obtain the following estimate for $T_{232}^{(1)}$:

Finally, for $T_{233}^{(1)}$ we have that

$$\begin{aligned} T_{233}^{(1)}&\le C_{233}^{(1)} N^{2(s-t)} \left( \frac{\epsilon }{\delta }\right) ^N \Vert q_2\Vert _{\delta R_2/\rho _2, t-1}^2 \sup _{n \in \varLambda _N' {\setminus } \{0\}} \left[ \left( \frac{N}{|n|}\right) ^{-2(s+1)} \left\{ \frac{1}{\epsilon \delta } \left( \frac{R_2}{\rho _2}\right) ^2 \right\} ^{N-2|n|} \right] . \end{aligned}$$

The above supremum can be bounded by some constant, because $(\epsilon \delta )^{-1} (R_2/\rho _2)^2 < 1$, and so we obtain that

$$\begin{aligned} T_{233}^{(1)} \le C_{233}^{(1)} N^{2(s-t)} \left( \frac{\epsilon }{\delta }\right) ^N \Vert q_2\Vert _{\delta R_2/\rho _2, t-1}^2. \end{aligned}$$

Summarizing the above, we obtain the following estimate for $T_2^{(1)}$:

Next, we give the estimate for $T_3^{(1)}$. We divide $T_3^{(1)}$ into two parts, as follows:

$$\begin{aligned} T_3^{(1)} = T_{31}^{(1)} + T_{32}^{(2)}, \end{aligned}$$

where

$$\begin{aligned} T_{31}^{(1)}&= \sum _{l \in \mathbb {Z}{\setminus } \{0\}} |lN|^{2(s-1)} \left( \frac{\epsilon \rho _1}{R_1}\right) ^{2|lN|} |\hat{q}_1^{(N)}(0)|^2, \\ T_{32}^{(2)}&= \sum _{p \in \varLambda _N' {\setminus } \{0\}} \left( \sum _{l \in \mathbb {Z}{\setminus } \{0\}} |p + lN|^{2(s-1)} \left( \frac{\epsilon \rho _1}{R_1}\right) ^{2|p+lN|} \right) |\hat{q}_1^{(N)}(p)|^2. \end{aligned}$$

First, we estimate $T_{31}^{(1)}$. By Proposition 8 (iv), we have that

$$\begin{aligned} \sum _{l \in \mathbb {Z}{\setminus } \{0\}} |lN|^{2(s-1)} \left( \frac{\epsilon \rho _1}{R_1}\right) ^{2|lN|}&= \sum _{m \in I(0)} |m|^{2(s-1)} \left( \frac{\epsilon \rho _1}{R_1}\right) ^{2|m|}\\&\le C_{\epsilon r, 2(s-1)} N^{2(s-1)} (\epsilon r)^{2N} \end{aligned}$$

and by Proposition 8 (i) and (ii) we have that

$$\begin{aligned} |\hat{q}_1^{(N)}(0)|^2&\le \frac{2}{(\det \varPhi ^\mathrm {C}(0))^2} \left[ \left( \sum _{m \equiv 0} \sum _{l \equiv 0} |\varUpsilon _1(m, l)| |\hat{q}_1(m)|\right) ^2\right. \\&\quad \left. + \left( \sum _{m \equiv 0} \sum _{l \equiv 0} |\varUpsilon _2(m, l)| |\hat{q}_2(m)|)\right) ^2 \right] \\&\le C \Vert q\Vert _{\mathbb {Y}_{\delta , t}}^2. \end{aligned}$$

Because it holds by definition that $\epsilon \le 1 / (\delta r^2)$, we obtain that

Concerning $T_{32}^{(1)}$, we have by virtue of Proposition 8 (iv) that

$$\begin{aligned} \sum _{l \in \mathbb {Z}{\setminus } \{0\}} |p + lN|^{2(s-1)} \left( \frac{\epsilon \rho _1}{R_1}\right) ^{2|p+lN|} \le C_{\epsilon \rho _1/R_1, 2(s-1)} N^{2(s-1)} \left( \frac{\epsilon \rho _1}{R_1}\right) ^{2(N-|p|)} \end{aligned}$$

for all $p \in \varLambda _N' {\setminus } \{0\}$. Furthermore, it follows from Proposition 8 (i) and (iii) that

$$\begin{aligned} |\hat{q}_1^{(N)}(p)|^2&\le \frac{2}{(\det \varPhi ^\mathrm {C}(p))^2} \left[ \left( \sum _{m \equiv p} \sum _{l \equiv p} |\varUpsilon _1(m, l)| |\hat{q}_1(m)|\right) ^2\right. \\&\quad \left. + \left( \sum _{m \equiv p} \sum _{l \equiv p} |\varUpsilon _2(m, l)| |\hat{q}_2(m)|\right) ^2 \right] \\&\le C |p|^2 \left( \frac{R_1}{\rho _1}\right) ^{2|p|} \left( \frac{1}{\delta ^{2|p|}} \frac{1}{|p|^{2t}} + \frac{1}{\delta ^{2(N-|p|)}} \frac{1}{N^{2t}} \right) \\&\quad \times \left[ \sum _{m \equiv p} |\hat{q}_1(m)|^2 \left( \frac{\delta \rho _1}{R_1}\right) ^{2|m|} \underline{m}^{2(t-1)}\right. \\&\quad \left. + \sum _{m \equiv p} |\hat{q}_2(m)|^2 \left( \frac{\delta R_2}{\rho _2}\right) ^{2|m|} \underline{m}^{2(t-1)} \right] . \end{aligned}$$

Therefore, we obtain that

$$\begin{aligned} T_{32}^{(1)} \le C_{32}^{(1)} N^{2[s - 1 + \max \{-t + 1, 0\}]} \left( \frac{\epsilon }{\delta }\right) ^N \Vert q\Vert _{\mathbb {Y}_{\delta , t}}^2 A_{32}^{(1)}, \end{aligned}$$

where

$$\begin{aligned} A_{32}^{(1)} = \sup _{p \in \varLambda _N' {\setminus } \{0\}} \left\{ \frac{N^{-2\max \{-t+1, 0\}}}{|p|^{2(t-1)}} (\epsilon \delta r^2)^{N - 2|p|} + |p|^2 N^{-2[t + \max \{-t+1, 0\}]} \left( \frac{\epsilon r^2}{\delta }\right) ^{N - 2|p|} \right\} . \end{aligned}$$

This supremum can be bounded as follows:

Then, we obtain the following estimate for $T_{32}^{(1)}$:

Summarizing the above, we have that

Finally, we will establish the estimate for $T_4^{(1)}$, which can be obtained by straightforward arguments as follows:

$$\begin{aligned} T_4^{(1)}&= \sum _{n \in \mathbb {Z}{\setminus } \varLambda _N'} |\hat{q}_1(n)|^2 \left( \frac{\delta \rho _1}{R_1}\right) ^{2|n|} \underline{n}^{2(t-1)} \cdot \frac{|n|^{2(s-1)}}{\underline{n}^{2(t-1)}} \left( \frac{\epsilon }{\delta }\right) ^{2|n|} \\&\le \frac{1}{(2\pi )^{2(t-1)}} N^{2(s-t)} \left( \frac{\epsilon }{\delta }\right) ^N \Vert q_1\Vert _{\delta \rho _1/R_1, t-1}^2 \sup _{n \in \mathbb {Z}{\setminus } \varLambda _N'} \left\{ \left( \frac{N}{|n|}\right) ^{-2(s-t)} \left( \frac{\epsilon }{\delta }\right) ^{2|n| - N} \right\} \\&\le C_4^{(1)} N^{2(s-t)} \left( \frac{\epsilon }{\delta }\right) ^N \Vert q_1\Vert _{\delta \rho _1/R_1, t-1}^2. \end{aligned}$$

Hence, we obtain the following estimate for $\Vert q_1 - q_1^{(N)}\Vert _{\epsilon \rho _1/R_1, s-1}^2$:

In addition, $\Vert q_2 - q_2^{(N)}\Vert _{\epsilon R_2/\rho _2, s-1}^2$ can be estimated in a similar manner. Thus, we obtain the desired estimate.

Proof of Lemma 2

We require one additional proposition as to the upper bound for $(\det \varPhi ^\mathrm {I}(0))^{-2}$, which will be used without proof.

Proposition 9

There exists some positive constant $C_{\rho _1,\,\rho _2,\,R_1,\,R_2}$ such that

$$\begin{aligned} \frac{1}{(\det \varPhi ^\mathrm {I}(0))^2} \le C_{\rho _1,\,\rho _2,\,R_1,\,R_2} \end{aligned}$$

holds for all $N \in \mathbb {N}$.

We can represent $\hat{q}_1^{(N)}(0)$ explicitly from (3.8), which yields that

$$\begin{aligned}&\hat{q}_1(0) - \hat{q}_1^{(N)}(0) \\&\quad = \frac{1}{\det \varPhi ^\mathrm {I}(0)} \left[ B_1 \hat{q}_1(0) - B_1 \hat{q}_2(0) + \sum _{l \in I(0)} \left( B_3(l) \hat{q}_1(l) + B_4(l) \hat{q}_2(l) \right) \right] , \end{aligned}$$

where

$$\begin{aligned} B_1 =&\frac{1}{2} \sum _{m \in I(0)} (-\hat{G}_{11}(m) + \hat{G}_{12}(m) + \hat{G}_{21}(m) - \hat{G}_{22}(m) - \varUpsilon _1(m, 0)\\&+ \varUpsilon _2(m, 0) - \varUpsilon _3(m, 0) + \varUpsilon _4(m, 0)), \\ B_3(l) =&\hat{G}_{11}(l) - \hat{G}_{12}(l) + \varUpsilon _4(0, l) - \varUpsilon _3(0, l) - \sum _{m \in I(0)} (\varUpsilon _3(m, l) - \varUpsilon _4(m, l)), \\ B_4(l) =&\hat{G}_{21}(l) - \hat{G}_{22}(l) - \varUpsilon _2(0, l) + \varUpsilon _1(0, l) - \sum _{m \in I(0)} (-\varUpsilon _1(m, l) + \varUpsilon _2(m, l)). \end{aligned}$$

Each term can be evaluated as follows, by Proposition 8 (i) and (iv):

$$\begin{aligned} |B_1 \hat{q}_1(0)|^2&\le C_{11}^{(1)} N^{-2} r^{2N} \Vert q_1\Vert _{\delta \rho _1/R_1, t-1}^2, \\ |B_1 \hat{q}_2(0)|^2&\le C_{12}^{(1)} N^{-2} r^{2N} \Vert q_2\Vert _{\delta R_2/\rho _2, t-1}^2,\\ \left| \sum _{l \in I(0)} B_3(l) \hat{q}_1(l) \right| ^2&\le C_{13}^{(1)} \sum _{l \in I(0)} \frac{1}{\delta ^{2|l|}} \frac{1}{\underline{l}^{2t}} \sum _{l \in I(0)} |\hat{q}_1(l)|^2 \left( \frac{\delta \rho _1}{R_1}\right) ^{2|l|} \underline{l}^{2t}\\&\le C_{13}^{(1)} N^{-2t} \delta ^{-2N} \Vert q_1\Vert _{\delta \rho _1/R_1, t-1}^2, \\ \left| \sum _{l \in I(0)} B_4(l) \hat{q}_2(l) \right| ^2&\le C_{14}^{(1)} \sum _{l \in I(0)} \frac{1}{\delta ^{2|l|}} \frac{1}{\underline{l}^{2t}} \sum _{l \in I(0)} |\hat{q}_2(l)|^2 \left( \frac{\delta R_2}{\rho _2}\right) ^{2|l|} \underline{l}^{2t} \\&\le C_{14}^{(1)} N^{-2t} \delta ^{-2N} \Vert q_2\Vert _{\delta R_2/\rho _2, t-1}^2. \end{aligned}$$

Therefore, using Proposition 9 (i), we obtain that

Because $\hat{q}_1^{(N)}(n)$ ($n \in \varLambda _N' {\setminus } \{0\}$) are the same as for C-MFS, we immediately obtain the estimate

Now we consider $T_3^{(1)}$, which we decompose as follows:

$$\begin{aligned} T_3^{(1)} = T_{31}^{(1)} + T_{32}^{(1)}, \end{aligned}$$

where

$$\begin{aligned} T_{31}^{(1)}&= \sum _{l \in \mathbb {Z}{\setminus } \{0\}} |lN|^{2(s-1)} \left( \frac{\epsilon \rho _1}{R_1}\right) ^{2|lN|} |\hat{q}_1^{(N)}(N)|^2, \\ T_{32}^{(2)}&= \sum _{p \in \varLambda _N' {\setminus } \{0\}} \left( \sum _{l \in \mathbb {Z}{\setminus } \{0\}} |p + lN|^{2(s - t)} \left( \frac{\epsilon \rho _1}{R_1}\right) ^{2|p + lN|} \right) |\hat{q}_1^{(N)}(p)|^2. \end{aligned}$$

The definition of $T_{31}^{(1)}$ is slightly different from that for C-MFS, while that of $T_{32}^{(1)}$ is the same as for C-MFS. Therefore, we only have to consider the estimate of $T_{31}^{(1)}$. From (3.8), we have that

$$\begin{aligned} \hat{q}_1^{(N)}(N)&= \frac{1}{\det \varPhi ^\mathrm {I}(0)} \left[ \frac{1}{2} (\hat{G}_{12}(0) - \hat{G}_{22}(0)) \hat{q}_1(0) + \frac{1}{2} (\hat{G}_{22}(0) - \hat{G}_{12}(0)) \hat{q}_2(0) \right. \\&\quad \left. +\, \sum _{l \in I(0)} \left( (\hat{G}_{21}(l) - \hat{G}_{11}(l)) \hat{q}_1(l) + (\hat{G}_{22}(l) - \hat{G}_{12}(l)) \hat{q}_2(l) \right) \right] , \end{aligned}$$

from which we can easily obtain by Proposition 8 (ii) and Proposition 9 that

$$\begin{aligned} |\hat{q}_1^{(N)}(N)|^2 \le C\Vert q\Vert _{\mathbb {Y}_{\delta , t}}^2. \end{aligned}$$

By using the results in Sect. 1, we have that

The estimate of $T_4^{(1)}$ is the same as for C-MFS. That is,

$$\begin{aligned} T_4^{(1)} \le C_4^{(1)} N^{2(s-t)} \left( \frac{\epsilon }{\delta }\right) ^N \Vert q_1\Vert _{\delta \rho _1/R_1, t-1}^2. \end{aligned}$$

Summarizing the above, we obtain the following estimate for $\Vert q_1 - q_1^{(N)}\Vert _{\epsilon \rho _1/R_1, s-1}^2$:

Then $\Vert q_2 - q_2^{(N)}\Vert _{\epsilon R_2/\rho _2, s-1}^2$ can be estimated in a similar manner, and we obtain the desired estimate.