Filtered interpolation for solving Prandtl’s integro-differential equations

Abstract

In order to solve Prandtl-type equations we propose a collocation-quadrature method based on de la Vallée Poussin (briefly VP) filtered interpolation at Chebyshev nodes. Uniform convergence and stability are proved in a couple of Hölder-Zygmund spaces of locally continuous functions. With respect to classical methods based on Lagrange interpolation at the same collocation nodes, we succeed in reproducing the optimal convergence rates of the L² case and cut off the typical log factor which seemed inevitable dealing with uniform norms. Such an improvement does not require a greater computational effort. In particular, we propose a fast algorithm based on the solution of a simple 2-bandwidth linear system and prove that, as its dimension tends to infinity, the sequence of the condition numbers (in any natural matrix norm) tends to a finite limit.

Appendix

1.1 Proof of Theorem 4.2

Firstly note that, by (55) and (30), we have

$$ \tilde {H_{n}^{m}}f(y)={\int}_{-1}^{1} \mathcal{H} (x,y)f(x)\varphi(x)dx, \qquad \mathcal{H}(x,y):=\sum\limits_{k=1}^{n} h(x_{k},y){\Phi}_{n,k}^{m}(x). $$

Let us prove that the boundedness of $\tilde {H_{n}^{m}}:C^{0}_{\varphi }\rightarrow Z_{s}(\varphi )$ follows from Proposition 2.3. Indeed, we recall that (cf. Theorem 3.2)

$$ \|{V_{n}^{m}}\|_{C^{0}_{\varphi}\rightarrow C^{0}_{\varphi}}:=\sup\limits_{\|f\varphi\|\le 1}\|({V_{n}^{m}}f)\varphi\|=\sup\limits_{|x|\le 1}\sum\limits_{k=1}^{n}\frac{|{\Phi}_{n,k}^{m}(x)|\varphi(x)}{\varphi(x_{k})}\le \mathcal{C}\ne\mathcal{C}(n,m). $$

(83)

Hence, from the assumptions on h, we deduce that

• (a) ${\mathscr{H}}(x,y)\varphi (x)\varphi (y)$ is a continuous function w.r.t. both the variables x, y ∈ [− 1, 1];

• (b) for all $n\in {\mathbb N}$ and |x|≤ 1, taking into account that

  $$ E_{n}(\overline{h}_{x_{k}})_{\varphi}= \|(\overline{h}_{x_{k}}-P^{*}_{x_{k}})\varphi\|\le \frac {\mathcal{C}}{n^{s}}, \quad k=1,\ldots, n, \qquad {\mathcal{C}}\ne{\mathcal{C}} (n,k), $$

  we set

  $$ P^{*}(y):=\sum\limits_{k=1}^{n} P^{*}_{x_{k}}(y){\Phi}_{n,k}^{m}(x)\varphi(x),\qquad |y|\le 1. $$

  Hence, we have

  $$ \begin{array}{@{}rcl@{}} E_{n}(\overline{\mathcal{H}}_{x}\varphi(x))_{\varphi}&=&\inf\limits_{P\in{\mathbb P}_{n}} \sup\limits_{|y|\le 1} \left| \mathcal{H}(x,y)\varphi(x)-P(y)\right|\varphi(y)\\ &\le&\sup\limits_{|y|\le 1} \left| \mathcal{H}(x,y)\varphi(x)-P^{*}(y)\right|\varphi(y)\\ &=& \sup\limits_{|y|\le 1}\left|\sum\limits_{k=1}^{n} \left[ h(x_{k},y)-P^{*}_{x_{k}}(y)\right]{\Phi}_{n,k}^{m}(x)\varphi(x)\right|\varphi(y)\\ &\le& \sum\limits_{k=1}^{n}\frac{|{\Phi}_{n,k}^{m}(x)|\varphi(x)}{\varphi(x_{k})} \sup\limits_{|y|\le 1}\left|\overline{h}_{x_{k}}(y)-P^{*}_{x_{k}}(y)\right|\varphi(y)\\ &\le& \frac C{n^{s}}\sum\limits_{k=1}^{n}\frac{|{\Phi}_{n,k}^{m}(x)|\varphi(x)}{\varphi(x_{k})} \le \frac {\mathcal{C}}{n^{s}}, \qquad {\mathcal{C}}\ne {\mathcal{C}}(n,x). \end{array} $$

In conclusion, by virtue of (a) and (b), the kernel ${\mathscr{H}}$ satisfies the assumptions of Proposition 2.3 with v = φ, so that the operator defined by this kernel, namely $\tilde {H_{n}^{m}}:C^{0}_{\varphi }\rightarrow Z_{s}(\varphi )$, is bounded.

Consequently, the map ${H_{n}^{m}}:C^{0}_{\varphi }\rightarrow Z_{s}(\varphi )$ is bounded too, since it is the following composition of bounded maps (cf. Theorem 3.2)

$$ H_{n}^{m}:C^{0}_{\varphi} \begin{array}{c} \tilde{H_{n}^{m}}\\ \longrightarrow \end{array} Z_{s}(\varphi) \begin{array}{c} {V_{n}^{m}}\\ \longrightarrow \end{array} Z_{s}(\varphi). $$

In order to prove (58), let us arbitrarily fix $f\in C^{0}_{\varphi }$ and |y|≤ 1.

By using (30), we note that

$$ {H_{n}^{m}}f(y)=\sum\limits_{j=1}^{n}\tilde{H_{n}^{m}}f(x_{j}){\Phi}_{n,j}^{m}(y)= {\int}_{-1}^{1}\left( \sum\limits_{j=1}^{n}\sum\limits_{k=1}^{n} h(x_{k},x_{j}){\Phi}_{n,k}^{m}(x){\Phi}_{n,j}^{m}(y)\right)f(x)\varphi(x)dx, $$

that means ${H_{n}^{m}}f$ can be obtained by replacing h with its bivariate VP interpolation polynomial based on tensor-product Chebyshev nodes of the second kind (see [23, 24]).

Hence, we write

$$ \begin{array}{@{}rcl@{}} Hf(y){-}{H_{n}^{m}}f(y)\!&=&\!{\int}_{-1}^{1}\left( h(x,y){-}\!\sum\limits_{j=1}^{n}h(x,x_{j}){\Phi}_{n,j}^{m}(y)\right)f(x)\varphi(x) dx\\ &+&\!\!{\int}_{-1}^{1}\sum\limits_{j=1}^{n}{\Phi}_{n,j}^{m}(y)\left( \!h(x,x_{j}){-}\!\! \sum\limits_{k=1}^{n}\!h(x_{k},x_{j}){\Phi}_{n,k}^{m}(x)\!\right)f(x)\varphi(x)dx. \end{array} $$

Then, using (30), (26) and (cf. (36))

$$ \sum\limits_{k=1}^{n}{\Phi}_{n,k}^{m}(x)=1,\qquad \forall |x|\le 1, $$

(84)

we get

$$ \begin{array}{@{}rcl@{}} Hf(y)-{H_{n}^{m}}f(y)&=& {\int}_{-1}^{1}\bigg(\overline{h}_{x}(y)-{V_{n}^{m}} (\overline{h}_{x})(y)\bigg)f(x)\varphi(x) dx\\ &+&{\int}_{-1}^{1}\sum\limits_{j=1}^{n}{\Phi}_{n,j}^{m}(y)\sum\limits_{k=1}^{n}{\Phi}_{n,k}^{m}(x) \bigg(\overline{h}_{x}(x_{j})-\overline{h}_{x_{k}}(x_{j})\bigg)f(x)\varphi(x)dx. \end{array} $$

Consequently

$$ \begin{array}{@{}rcl@{}} &&|Hf(y)-{H_{n}^{m}}f(y)|\varphi(y)\le{\int}_{-1}^{1}\bigg|\overline{h}_{x}(y)-{V_{n}^{m}} (\overline{h}_{x})(y)\bigg| \varphi(y)|f(x)|\varphi(x) dx\\ &+&\sum\limits_{j=1}^{n} |{\Phi}_{n,j}^{m}(y)|\varphi(y){\int}_{-1}^{1}\sum\limits_{k=1}^{n}|{\Phi}_{n,k}^{m}(x)| \bigg|\overline{h}_{x}(x_{j})-\overline{h}_{x_{k}}(x_{j})\bigg| |f(x)|\varphi(x)dx\\ &=:&A+B. \end{array} $$

On the other hand, the hypothesis $\overline {h}_{x}\in Z_{s}(\varphi )$ uniformly w.r.t. x ∈ [− 1, 1] implies that $\forall n\in {\mathbb N}$ and ∀|x|≤ 1 we have

$$ E_{n}(\overline{h}_{x})_{\varphi}\le \frac {\mathcal{C}}{n^{s}},\qquad {\mathcal{C}}\ne {\mathcal{C}}(n,x), $$

(85)

and in particular, there exists $P^{*}\in {\mathbb P}_{n}$ such that

$$ \sup\limits_{|y|\le 1}\bigg|\overline{h}_{x}(y)-P^{*}(y)\bigg|\varphi(y)\le \frac {\mathcal{C}}{n^{s}},\qquad {\mathcal{C}}\ne {\mathcal{C}}(n,x). $$

(86)

Hence, concerning A, by (48) and (85), recalling that m = ⌊𝜃n⌋ with a fixed 0 < 𝜃 < 1, we get

$$ A\le {\int}_{-1}^{1} E_{n-m}(\overline{h}_{x})_{\varphi} |f(x)|\varphi(x) dx\le \frac{\mathcal{C}}{(n-m)^{s}} {\int}_{-1}^{1} |f(x)|\varphi(x) dx\le \frac{\mathcal{C}}{n^{s}}\|f\varphi\|. $$

Moreover, regarding B, by (86) and (83), we have

$$ \begin{array}{@{}rcl@{}} B&\le&\|f\varphi\| \sum\limits_{j=1}^{n}\frac{|{\Phi}_{n,j}^{m}(y)|\varphi(y)}{\varphi(x_{j})} {\int}_{-1}^{1}\sum\limits_{k=1}^{n}\frac{|{\Phi}_{n,k}^{m}(x)|\varphi(x)}{\varphi(x_{k})} \bigg|(\overline{h}_{x}-P^{*})(x_{j}){-} (\overline{h}_{x_{k}}-P^{*})(x_{j})\bigg| \frac{\varphi(x_{j})}{\varphi(x)}dx\\ &\le&\frac {\mathcal{C}}{n^{s}} \|f\varphi\| \sum\limits_{j=1}^{n}\frac{|{\Phi}_{n,j}^{m}(y)|\varphi(y)}{\varphi(x_{j})} {\int}_{-1}^{1}\sum\limits_{k=1}^{n}\frac{|{\Phi}_{n,k}^{m}(x)|\varphi(x)}{\varphi(x_{k})} \frac{dx}{\varphi(x)}\\ &\le& \frac {\mathcal{C}}{n^{s}} \|f\varphi\|{\int}_{-1}^{1} \frac{dx}{\varphi(x)}\le\frac {\mathcal{C}}{n^{s}}\|f\varphi\|, \qquad\text{with} {\mathcal{C}}\ne{\mathcal{C}}(n,f,y). \end{array} $$

Finally, let us prove (57). Taking into account that ${H_{n}^{m}}f\in {\mathbb P}_{n+m-1}$ and recalling that $H:C^{0}_{\varphi }\rightarrow Z_{s}(\varphi )$ is bounded (cf. Proposition 2.3), for any $f\in C^{0}_{\varphi }$ we note that

$$ \begin{array}{@{}rcl@{}}\sup\limits_{k\ge n+m-1} (k+1)^{r} E_{k}(Hf-{H_{n}^{m}}f)_{\varphi}&=&\sup\limits_{k\ge n+m-1} (k+1)^{r} E_{k}(Hf)_{\varphi}\\ &\le& {\mathcal{C}} \sup\limits_{k\ge n+m-1} \frac{\|Hf\|_{Z_{s}(\varphi)}}{k^{s-r}}\le {\mathcal{C}}\frac{\|f\varphi\|}{n^{s-r}}. \end{array} $$

Moreover, by (58) we get

$$ \begin{array}{@{}rcl@{}} \sup\limits_{k< n+m-1} (k+1)^{r} E_{k}(Hf-{H_{n}^{m}}f)_{\varphi}&\le& \|(Hf-{H_{n}^{m}}f)\varphi\|\sup\limits_{k< n+m-1} (k+1)^{r} \\ &\le& {\mathcal{C}} \frac{\|f\varphi\|}{n^{s}}(n+m)^{r}\le {\mathcal{C}}\frac{\|f\varphi\|}{n^{s-r}}. \end{array} $$

Consequently, by (58) we have

$$ \|(H-{H_{n}^{m}})f \|_{Z_{r}(\varphi)}=\| (Hf-{H_{n}^{m}}f)\varphi \| + \sup\limits_{k\in{\mathbb N}} (k+1)^{r} E_{k}(Hf-{H_{n}^{m}}f)_{\varphi}\le {\mathcal{C}}\frac{\|f\varphi\|}{n^{s-r}} $$

and (57) follows.

1.2 Proof of Theorem 4.4

Note that we can write

$$ \begin{array}{@{}rcl@{}} f^{*}-\tilde {f_{n}^{m}}&=&(D+{U_{n}^{m}})^{-1}\left[(D+{U_{n}^{m}})f^{*}- {V_{n}^{m}}g \right]\\ &=&(D+{U_{n}^{m}})^{-1}\left[(g- {V_{n}^{m}}g)+ ({U_{n}^{m}}-U)f^{*}\right]. \end{array} $$

Hence, by the uniform boundedness of $(D+{U_{n}^{m}})^{-1}:Z_{r}(\varphi )\rightarrow Z_{r+1}(\varphi )$ (cf. Th. 4.3), for any 0 < r ≤ s we deduce

$$ \|f^{*}-\tilde {f_{n}^{m}}\|_{Z_{r+1}(\varphi)}\le {\mathcal{C}} \left[\|g- {V_{n}^{m}}g\|_{Z_{r}(\varphi)}+ \|({U_{n}^{m}}-U)f^{*}\|_{Z_{r}(\varphi)}\right] $$

and (62) follows from Th. 3.2, Th. 4.1 and Th. 4.2, taking into account that, by hypothesis, g ∈ Z_s(φ), f^∗∈ Z_s+ 1(φ) and $\|g\|_{Z_{s}(\varphi )}\sim \|f^{*}\|_{Z_{s+1}(\varphi )}$.

Finally, (63) follows from (62) when $r\rightarrow 0^{+}$.

1.3 Proof of Lemma 5.1

Let $\displaystyle \tilde f(y)=\sum\limits_{j=0}^{n-1}\tilde f_{j} \tilde q_{j}(y)$. Recalling (41)–(42), we have $\displaystyle {V_{n}^{m}}\tilde f(y)= \sum \limits_{r=0}^{n-1}q_{r}(y)c_{n,r}(\tilde f),$ where

$$ \begin{array}{@{}rcl@{}} c_{n,r}(\tilde f)&=&\sum\limits_{k=1}^{n}\lambda_{k} p_{r}(x_{k})\tilde f(x_{k}) \\ &=&\left( \sum\limits_{j=0}^{n-m}+ \sum\limits_{j=n-m+1}^{n-1}\right) \tilde f_{j} \sum\limits_{k=1}^{n}\lambda_{k} p_{r}(x_{k})\widetilde q_{j}(x_{k})\\ &=& \sum\limits_{j=0}^{n-m} \tilde f_{j} \sum\limits_{k=1}^{n}\lambda_{k} p_{r}(x_{k})\frac{p_{j}(x_{k})}{j+1}\\ &+& \sum\limits_{j=n-m+1}^{n -1} \tilde f_{j} \sum\limits_{k=1}^{n}\lambda_{k} p_{r}(x_{k})\left( \frac{m+n-j}{2m(j+1)}p_{j}(x_{k})- \frac{(j-n+m)}{2m(2n-j+1)}p_{2n-j}(x_{k})\right) . \end{array} $$

Consequently, by using [30]

$$ p_{2n-j}(x_{k})=- p_{j}(x_{k}), \qquad k=1,\ldots, n,\qquad n-m<j<n, $$

(87)

for any r = 0,…,n − 1, we get

$$ \begin{array}{@{}rcl@{}} c_{n,r}(\tilde f)&=&\sum\limits_{j=0}^{n-m}\tilde f_{j} < p_{r},p_{j}>\frac 1 {j+1}\\ &+& \sum\limits_{j=n-m+1}^{n -1}\tilde f_{j} < p_{r},p_{j}>\frac{m+n-j}{2m(j+1)}+ \sum\limits_{j=n-m+1}^{n -1}\tilde f_{j} <p_{r},p_{j}>\frac{(j-n+m)}{2m(2n-j+1)}. \end{array} $$

Hence, the orthogonality relation < p_r,p_j >= δ_{r, j} implies that

$${V_{n}^{m}}\tilde f(y)= \sum\limits_{r=0}^{n-m}q_{r}(y)\tilde f_{r}\left[\frac{1}{r+1}\right]+ \sum\limits_{r=n-m+1}^{n-1}q_{r}(y)\tilde f_{r}\left[\frac 1{2m} \left( \frac{m+n-r}{r+1}+\frac{r-n+m}{2n-r+1}\right)\right], $$

and the statement follows.

1.4 Proof of Lemma 5.2

Due to the assumption $\displaystyle \tilde f(y)= \sum\limits_{\ell =0}^{n-1}\tilde f_{\ell }\ \tilde q_{\ell }(y)$, the core of the proof consists in stating that

$$ K \widetilde q_{\ell}(x_{k})= \alpha_{\ell} p_{\ell-2}(x_{k})+\beta_{\ell} p_{\ell}(x_{k})+\gamma_{\ell} p_{\ell+2}(x_{k}),\quad \ell=0,1,\dots,n -1. $$

(88)

This is an immediate consequence of (43), (22) and (23) in the case that ℓ = 0,…, (n − m). In order to prove (88) also in the case (n − m) < ℓ < n, we observe that in this case the previous equations yield

$$ \begin{array}{@{}rcl@{}} K \widetilde q_{\ell}(x_{k})&=& \frac{n+m-\ell}{2m(\ell+1)}K p_{\ell}(x_{k})-\frac{\ell-n+m}{2m(2n-\ell+1)}K p_{2 n-\ell}(x_{k})\\ &=& \frac{n+m-\ell}{8m(\ell+1)}\left[-\frac 1 \ell p_{\ell-2}(x_{k})+\left( \frac 1 \ell+\frac 1 {\ell+2}\right)p_{\ell}(x_{k})-\frac 1 {\ell+2}p_{\ell+2}(x_{k}) \right]\\ &-&\frac{\ell-n+m}{8m(2n-\ell+1)}\left[-\frac 1 {2n-\ell} p_{2n-\ell-2}(x_{k})+\left( \frac 1 {2n-\ell}+\frac 1 {2n-\ell+2}\right)p_{2n-\ell}(x_{k})\right. \\ &-& \left.\frac 1 {2n-\ell+2}p_{2n-\ell+2}(x_{k})\right], \end{array} $$

and using (87), we get

$$ \begin{array}{@{}rcl@{}} K \widetilde q_{\ell}(x_{k})= &-&\left[\frac{n+m-\ell}{8m\ell(\ell+1)}+\frac{\ell-n+m}{8m(2n-\ell+1)(2n-\ell+2)}\right] p_{\ell-2}(x_{k}) \\ &+&\left[\frac{n+m-\ell}{4m\ell(\ell+2)}+\frac{\ell-n+m}{4m(2n-\ell)(2n-\ell+2)}\right]p_{\ell}(x_{k})\\ & - &\left[ \frac{n+m-\ell}{8m(\ell+1)(\ell+2)}+\frac{\ell-n+m}{8m(2n-\ell+1)(2n-\ell)} \right]p_{\ell+2}(x_{k}) .\end{array} $$

(89)

By (89), the statement follows if (n − m) < ℓ ≤ (n − 3) and the same holds if ℓ = (n − 2), being in this case p_ℓ+ 2(x_k) = p_n(x_k) = 0.

If ℓ = (n − 1) then (89) implies that

$$ \begin{array}{@{}rcl@{}} K \widetilde q_{n-1}(x_{k})= &-&\left[\frac{m+1}{8 m n (n-1)}+\frac{m-1}{8m(n+2)(n+3)}\right]p_{n-3}(x_{k}) \\ & +&\left[\frac{m+1}{4m(n-1)(n+1)}+\frac{m-1}{4m(n+1)(n+3)}\right]p_{n -1}(x_{k})\\ & - &\left[\frac{m+1}{8 m n (n+1)}+\frac{m-1}{8 m(n+2)(n+1)}\right]p_{n+1}(x_{k}), \end{array} $$

and taking into account that p_n− 1(x_k) = −p_n+ 1(x_k) (cf. (87)), we get

$$ \begin{array}{@{}rcl@{}} K \widetilde q_{n-1}(x_{k})= &-&\left[\frac{m+1}{8 m n (n-1)}+\frac{m-1}{8m(n+2)(n+3)}\right]p_{n-3}(x_{k}) \\ &+&\left[\frac{(m+1)(3n-1)}{8 m n (n-1)(n+1)}+\frac{(m-1)(3n+7)}{8m(n+1)(n+2)(n+3)}\right]p_{n-1}(x_{k}), \end{array} $$

which concludes the proof of (88).

Finally, by (88), we get the statement as follows

$$ \begin{array}{@{}rcl@{}} {K_{n}^{m}}\tilde f (y)&=&\sum\limits_{j=0}^{n-1}q_{j}(y)\left[\sum\limits_{k=1}^{n}\lambda_{k} p_{j}(x_{k})\sum\limits_{\ell=0}^{n-1} \tilde f_{\ell} K \widetilde q_{\ell}(x_{k})\right] \\ &=&\sum\limits_{j=0}^{n-1}q_{j}(y)\left[\sum\limits_{k=1}^{n}\lambda_{k} p_{j}(x_{k})\sum\limits_{\ell=0}^{n-1}\tilde f_{\ell} \bigg(\alpha_{\ell} p_{\ell-2}(x_{k})+\beta_{\ell} p_{\ell}(x_{k})+\gamma_{\ell} p_{\ell+2}(x_{k})\bigg)\right]\\ & =& \sum\limits_{j=0}^{n-1}q_{j}(y)\sum\limits_{\ell=0}^{n-1}\tilde f_{\ell} \bigg[\alpha_{\ell} <p_{\ell-2}, p_{j}>+\beta_{\ell}<p_{\ell}, p_{j}>+ \gamma_{\ell}<p_{\ell+2}, p_{j}>\bigg]\\ &=&\sum\limits_{j=0}^{n-1}q_{j}(y)\bigg[ \alpha_{j+2}\tilde f_{j+2}+\beta_{j}\tilde f_{j}+\gamma_{j-2}\tilde f_{j-2}\bigg]. \end{array} $$

1.5 Proof of Lemma 5.3

The statement can be deduced from (54)–(56) and (44), as follows

$$ \begin{array}{@{}rcl@{}} {H_{n}^{m}} \tilde f(y)&=&\sum\limits_{j=0}^{n-1}q_{j}(y)\left[\sum\limits_{k=1}^{n}\lambda_{k} p_{j}(x_{k})\widetilde H_{n} \tilde f(x_{k})\right] \\ &=&\frac 1 \pi\sum\limits_{j=0}^{n-1}q_{j}(y)\sum\limits_{k=1}^{n}\lambda_{k} p_{j}(x_{k})\left[\sum\limits_{i=0}^{n-1}\sum\limits_{s=1}^{n}\lambda_{s} p_{i}(x_{s})h(x_{s},x_{k}){\int}_{-1}^{1}\tilde f(x) q_{i}(x)\varphi(x)dx\right]\\ &=&\frac 1 \pi\sum\limits_{j=0}^{n-1}q_{j}(y)\left[\sum\limits_{i=0}^{n-1}\sum\limits_{k=1}^{n} \sum\limits_{s=1}^{n}\lambda_{k} \lambda_{s} p_{j}(x_{k})p_{i}(x_{s})h(x_{s},x_{k})\sum\limits_{\ell=0}^{n-1} \tilde f_{\ell}<\widetilde q_{\ell}, q_{i}>\right]\\ &=& \frac 1 \pi\sum\limits_{j=0}^{n-1}q_{j}(y)\left[\sum\limits_{i=0}^{n-1}\sum\limits_{k=1}^{n} \sum\limits_{s=1}^{n}\lambda_{k} \lambda_{s} p_{j}(x_{k})p_{i}(x_{s})h(x_{s},x_{k})\tilde f_{i}<\widetilde q_{i}, q_{i}>\right]. \end{array} $$

1.6 Proof of Theorem 5.4

Firstly, let us focus on the elements α_k,γ_k,β_k and w_k that have been introduced by Lemmas 5.2 and 5.1, respectively. We note that they are decreasing in modulus and tend to 0 as $n\to \infty$, |α_i|,|γ_i|,β_i with order 1/n² and w_i with order 1/n. Consequently, for any matrix norm ∥⋅∥ and for any ε > 0, there exists ν_ε such that the matrix $\mathcal {A}^{\prime }_{n}$ can be splitted into the following sum

$$ \mathcal{A}^{\prime}_{n}=\mathcal{R}_{n}+\mathcal{E}_{n}, \qquad \text{with}\ \|\mathcal{E}_{n}\|< \varepsilon, \quad\forall n>\nu_{\epsilon}. $$

(90)

More precisely, for any n > ν_ε we represent the matrix $\mathcal {A}^{\prime }_{n}$ in the following block form

$$ \mathcal{A}^{\prime}_{n} =: \begin{pmatrix}A_{1,1} & A_{1,2}\\ A_{2,1} & A_{2,2}\end{pmatrix},$$

where

$$ \begin{array}{@{}rcl@{}}A_{1,1}&:=&\begin{pmatrix} \delta_{0} & 0 & \alpha_{2} & & & & \\ 0 & \delta_{1} & 0 & \alpha_{3} & & \text{ 0} \\ \gamma_{0} & 0 & \delta_{2} & 0 & {\ddots} & \\ & {\ddots} & {\ddots} & {\ddots} & {\ddots} & \alpha_{\nu_{\varepsilon}} \\ &\text{ 0} & {\ddots} & {\ddots} & {\ddots} & 0\\ & & & \gamma_{\nu_{\varepsilon}-2} & 0 & \delta_{\nu_{\varepsilon}}\\ \end{pmatrix}\in {\mathbb R}^{(\nu_{\varepsilon}+1)\times (\nu_{\varepsilon}+1)} \\ [.15in] A_{1,2}&:=&\begin{pmatrix} 0 & 0 & {\ldots} & 0 & 0 \\ {\vdots} & 0 & 0 & 0 & 0 \\ 0 & {\vdots} & 0 & {\vdots} & \vdots\\ \alpha_{\nu_{\varepsilon}+1} & 0 & {\vdots} & 0 & 0 \\ 0 & \alpha_{\nu_{\varepsilon}+2} & 0 & {\ldots} & 0 \end{pmatrix} \in {\mathbb R}^{(\nu_{\varepsilon}+1)\times (n-\nu_{\varepsilon}-1)} \\ [.15in] A_{2,1}&:=& \begin{pmatrix} 0 & 0 & 0 & \gamma_{\nu_{\varepsilon}-1} & 0\\ 0 & {\vdots} & {\vdots} & 0 & \gamma_{\nu_{\varepsilon}} \\ {\vdots} & {\vdots} & 0 & {\ldots} & 0\\ 0 & 0 & 0 & {\vdots} & 0 & \\ 0 & 0 & {\ldots} & 0 & 0 \end{pmatrix} \in {\mathbb R}^{(n-\nu_{\varepsilon}-1)\times (\nu_{\varepsilon}+1) } \end{array} $$

$$ \begin{array}{@{}rcl@{}} \\ [.15in] A_{2,2}&:=&\begin{pmatrix} \delta_{\nu_{\varepsilon}+1} & 0 & \alpha_{\nu_{\varepsilon}+3} & & & & \\ 0 & \delta_{\nu_{\varepsilon}+2} & 0 & \alpha_{\nu_{\varepsilon}+4} & &\text{ 0} & \\ \gamma_{\nu_{\varepsilon}+1} & 0 & \delta_{\nu_{\varepsilon}+3} & 0 & & & \\ & & {\ddots} & {\ddots} & {\ddots} & & \\ & & {\ddots} & {\ddots} & {\ddots} & & \alpha_{n}\\ & \text{ 0} & & {\ddots} & {\ddots} & & 0 \\ & & & & \gamma_{n-2} & 0 & \delta_{n-1}\\ \end{pmatrix}\in {\mathbb R}^{(n-\nu_{\varepsilon}-1)\times (n-\nu_{\varepsilon}-1)} \end{array} $$

and we recall that δ_k = 1 + σw_k + β_k has been defined in (80).

Hence, for a suitable choice of ν_𝜖 (depending on the choice of the matrix norm and 𝜖 > 0) we get (90) by taking

$$ \begin{array}{@{}rcl@{}} \mathcal{R}_{n}&:=&\begin{pmatrix}A_{1,1} & \mathbf{O}_{(\nu_{\varepsilon}+1)\times (n-\nu_{\varepsilon}-1)}\\ \mathbf{O}_{(n-\nu_{\varepsilon}-1)\times (\nu_{\varepsilon}+1)}& \mathcal{I}_{n-\nu_{\varepsilon}-1} \end{pmatrix}, \\ [.1in] \mathcal{E}_{n}&:=&\begin{pmatrix}\mathbf{O}_{(\nu_{\varepsilon}+1)\times (\nu_{\varepsilon}+1)} & A_{1,2}\\ A_{2,1} & A_{2,2}-\mathcal{I}_{n-\nu_{\varepsilon}-1} \end{pmatrix}, \end{array} $$

where O_k×h denotes the null matrix in ${\mathbb R}^{k\times h}$ and $\mathcal {I}_{k}$ denotes the identity k × k matrix.

From (90), using $\|\mathcal {E}_{n}\|<\varepsilon$, we easily deduce

$$ \|\mathcal{R}_{n}\|-\varepsilon\le \| \mathcal{A}^{\prime}_{n}\|\le \|\mathcal{R}_{n}\|+\varepsilon, \quad \forall \epsilon>0, \quad \forall n>\nu_{\epsilon}, $$

(91)

where we remark that $\|\mathcal {R}_{n}\|$ is independent of n > ν_ε, since increasing the order n of $\mathcal {R}_{n}$, the order of the identity block increases, while A_1,1 remains unchanged. Consequently, by (91) we get

$$\forall \varepsilon>0 \ \exists \nu_{\varepsilon}:\ \forall n_{1},n_{2}>\nu_{\varepsilon}, \ \left|\|\mathcal{A}^{\prime}_{n_{1}}\|-\|\mathcal{A}^{\prime}_{n_{2}}\|\right|< 2 \varepsilon,$$

i.e., the sequence $\{\|\mathcal {A}^{\prime }_{n}\|\}_{n}$ converges since it is a Cauchy sequence.

In order to complete the proof, let us prove that also the sequence $\{\|({\mathcal {A}^{\prime }}_{n})^{-1}\|\}_{n}$ is a Cauchy sequence.

Starting again from (90), we observe the matrix $\mathcal {R}_{n}$ is nonsingular and, under the previous settings, we have

$$ \mathcal{R}_{n}^{-1}=\begin{pmatrix}A_{1,1}^{-1} & \mathbf{O}\\ \mathbf{O} & \mathcal{I} \end{pmatrix}, $$

(92)

where, for the sake of brevity, we omit to explicit the dimensions of the null and identity blocks.

Consequently, by (90) we get

$$\mathcal{A}^{\prime}_{n}=\mathcal{R}_{n}(\mathcal{I}_{n}+\mathcal{R}_{n}^{-1}\mathcal{E}_{n}).$$

Let us prove that ν_𝜖 can be chosen in such a way that also $\|\mathcal {R}_{n}^{-1}\mathcal {E}_{n}\|<\epsilon$ holds true for any n > ν_𝜖. Indeed, we note that

$$ \mathcal{R}_{n}^{-1}\mathcal{E}_{n} =\begin{pmatrix} \mathbf{O} & A_{1,1}^{-1}A_{1,2}\\ A_{2,1} & A_{22}- \mathcal{I}\end{pmatrix} $$

(93)

differs from $\mathcal {E}_{n}$ only for the block matrix in the position (1,2), which is $A_{1,1}^{-1}A_{1,2}$ instead of A_1,2. Nevertheless, in terms of the column vectors, denoted by $\mathbf {e}_{i}\in {\mathbb R}^{\nu _{\varepsilon }+1}$ the i − th vector of the canonical basis of ${\mathbb R}^{\nu _{\varepsilon }+1}$ and by $\mathbf {0}\in {\mathbb R}^{\nu _{\epsilon } +1}$ the null vector, we note that

$$ \begin{array}{@{}rcl@{}} A_{1,2}&=&[\alpha_{\nu_{\varepsilon}+1} \mathbf{e}_{\nu_{\varepsilon}},\ \alpha_{\nu_{\varepsilon}+2} \mathbf{e}_{\nu_{\varepsilon}+1}, \ \mathbf{0}, \ldots, \mathbf{0}], \\ A_{1,1}^{-1}A_{1,2}&=&[\alpha_{\nu_{\varepsilon}+1} \mathbf{P}^{0}_{\nu_{\varepsilon}},\ \alpha_{\nu_{\varepsilon}+2} \mathbf{P}^{1}_{\nu_{\varepsilon}+1}, \ \mathbf{0}, \ldots, \mathbf{0}], \end{array} $$

where we set

$$\mathbf{P}^{0}:=A_{1,1}^{-1}\cdot \mathbf{e}_{\nu_{\varepsilon}}, \quad \mathbf{P}^{1}:=A_{1,1}^{-1}\cdot \mathbf{e}_{\nu_{\varepsilon}+1}.$$

In order to obtain an explicit expression of these vectors, we need only the last two columns of $A_{1,1}^{-1}$, which we deduce from the following LU factorization of A_1,1

$$A_{1,1}=LU=\begin{pmatrix} 1 & & & & & & \\ 0 & 1 & & & & \textrm{ 0} & \\ v_{0} & 0 & 1 & & & & \\ & v_{1} & 0 & 1 & & & \\ & & {\ddots} & {\ddots} & {\ddots} & & \\ & \textrm{ 0} & & v_{\nu_{\varepsilon}-2} & 0 & 1 & \end{pmatrix} \begin{pmatrix} d_{0} & 0 & \alpha_{2} & & & & \\ & d_{1} & 0 & \alpha_{3} & & \textrm{ 0} & \\ & & d_{2} & 0 & {\ddots} & & \\ & & & {\ddots} & {\ddots} & \alpha_{\nu_{\varepsilon}} \\ & & \textrm{ 0} & & d_{\nu_{\epsilon}-1}& 0 \\ & & & & & d_{\nu_{\varepsilon}} \end{pmatrix}, $$

where d₀ = δ₀, d₁ = δ₁, and for $k=2,3,{\dots } \nu _{\varepsilon }$, we have

$$ \begin{array}{@{}rcl@{}}\ v_{k-2}&:=&\frac{\gamma_{k-2}}{d_{k-2}},\\ d_{k}&:=&\delta_{k}-v_{k-2}\alpha_{k}= 1+\sigma w_{k}+\beta_{k}-v_{k-2}\alpha_{k}.\end{array} $$

More precisely, it can be checked that the elements $\mathbf {P}^{0}_{i}$ and $\mathbf {P}^{1}_{i}$, i = 0,…,ν_𝜖, of the previous vectors P⁰ and P¹ can be determined by induction as follows

$$ \begin{array}{@{}rcl@{}} \mathbf{P}^0_{0}&=&\frac{1-\alpha_2 \mathbf{P}^0_2}{d_0},\\ \mathbf{P}^0_{2i+1}&=&0, \quad i=0,1,\ldots,\frac{\nu_\varepsilon}2-1,\\ \mathbf{P}^0_{2i}&=& \frac{(-1)^i{\prod}_{j=0}^{i-1}v_{2j}-\alpha_{2i+2}\mathbf{P}^0_{2i+2}}{d_{2i}} \quad i=\frac{\nu_\varepsilon}{2}-1, \frac{\nu_\varepsilon}{2}-2,\ldots,1,\\ \mathbf{P}^0_{\nu_\varepsilon}&=& \frac{(-1)^{\frac{\nu_\varepsilon}{2}}{\prod}_{j=0}^{\frac{\nu_\varepsilon}{2}-1}v_{2j}}{d_{\nu_\varepsilon}},\\ \mathbf{P}^1_{1}&=& \frac{1-\alpha_{3}\mathbf{P}^1_{3}}{d_{1}},\\ \mathbf{P}^1_{2i+1}&=& \frac{(-1)^{i}{\prod}_{j=0}^{i-1}v_{2j+1}-\alpha_{2i+3}\mathbf{P}^{1}_{2i+3}}{d_{2i+1}} \quad i=\frac{\nu_\varepsilon}{2}-3, \ldots,2,\\ \quad \mathbf{P}^1_{2i} & = & 0, \quad i=0,1,\ldots,\frac{\nu_\varepsilon}{2},\\ \mathbf{P}^{1}_{\nu_\varepsilon-1} & = & \frac{(-1)^{\frac{\nu_\varepsilon}{2}-1}{\prod}_{j=0}^{\frac{\nu_\varepsilon}{2}-2}v_{2j+1}}{d_{\nu_\varepsilon-1}}. \end{array} $$

Thus, recalling the behavior of the sequences α_k,β_k,γ_k,w_k, we can conclude that, under a suitable choice of ν_ε, all the entries of $\mathcal {R}_{n}^{-1}\mathcal {E}_{n}$ are so small that we have

$$ \mathcal{A}^{\prime}_{n}=\mathcal{R}_{n}(\mathcal{I}_{n}+\mathcal{R}_{n}^{-1}\mathcal{E}_{n}), \qquad\text{with }\ \|\mathcal{R}_{n}^{-1}\mathcal{E}_{n}\|<\varepsilon, \qquad \forall n>\nu_{\epsilon} . $$

Consequently, taking 0 < 𝜖 < 1 and recalling that $\|(\mathcal {I}_{n}+\mathcal {R}_{n}^{-1}\mathcal {E}_{n})^{-1}\|\le \frac 1 {1-\|\mathcal {R}_{n}^{-1}\mathcal {E}_{n}\|}$ (see, e.g., [13, Lemma 2.3.3, p. 59]) we have

$$ \|{(\mathcal{A}^{\prime}}_{n})^{-1}\|= \|(\mathcal{I}_{n}+\mathcal{R}_{n}^{-1}\mathcal{E}_{n})^{-1}\mathcal{R}_{n}^{-1}\| \le \frac{\|\mathcal{R}_{n}^{-1}\|}{1-\varepsilon},\qquad \forall n>\nu_{\varepsilon}, $$

as well as, we get

$$ \|\mathcal{R}_{n}^{-1}\|= \|(\mathcal{I}_{n}+\mathcal{R}_{n}^{-1}\mathcal{E}_{n})(\mathcal{A}^{\prime}_{n})^{-1}\|\le (1+\epsilon)\|(\mathcal{A}^{\prime}_{n})^{-1}\|,\qquad \forall n>\nu_{\varepsilon}. $$

Summing up, we have stated that

$$ \frac{\|\mathcal{R}_{n}^{-1}\|}{1+\varepsilon}\le \|{(\mathcal{A}^{\prime}}_{n})^{-1}\|\le \frac{\|\mathcal{R}_{n}^{-1}\|}{1-\varepsilon},\qquad \forall n>\nu_{\varepsilon}, \quad 0<\epsilon<1 . $$

(94)

On the other hand, in view of (92), we can say that $\|\mathcal {R}_{n}^{-1}\|$ is independent of n > ν_𝜖, like $\|\mathcal {R}_{n}\|$, and in particular there exists a constant M independent of n s.t. $\|\mathcal {R}_{n}^{-1}\|\le M$. Hence, by (94) we get for any n₁,n₂ > ν_ε and 0 < ε < 1

$$ \left|\|({\mathcal{A}^{\prime}}_{n_{1}})^{-1}\|-\|({\mathcal{A}^{\prime}}_{n_{2}})^{-1}\|\right|\le \frac{2\varepsilon}{1-\varepsilon^{2}}\|\mathcal{R}_{n}^{-1}\|\le \frac{2M\varepsilon}{1-\varepsilon^{2}},$$

by which

$$\lim_{n_{1},n_{2}\to\infty}\left|\|({\mathcal{A}^{\prime}}_{n_{1}})^{-1}\| -\|({\mathcal{A}^{\prime}}_{n_{2}})^{-1}\|\right|=0,$$

and the statement follows.