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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.

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Appendix

Appendix

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

  1. (a)

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

  2. (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.

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 < rs 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, gZs(φ), fZs+ 1(φ) and \(\|g\|_{Z_{s}(\varphi )}\sim \|f^{*}\|_{Z_{s+1}(\varphi )}\).

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

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 < pr,pj >= δr, 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.

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 = 0,…, (nm). In order to prove (88) also in the case (nm) < < 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 (nm) < ≤ (n − 3) and the same holds if = (n − 2), being in this case p+ 2(xk) = pn(xk) = 0.

If = (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 pn− 1(xk) = −pn+ 1(xk) (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

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

Proof of Lemma 5.3

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

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

Proof of Theorem 5.4

Firstly, let us focus on the elements αk,γk,βk and wk 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 \(n\to \infty \), |αi|,|γi|,βi with order 1/n2 and wi with order 1/n. Consequently, for any matrix norm ∥⋅∥ and for any ε > 0, there exists νε 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 > νε 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 δk = 1 + σwk + βk has been defined in (80).

Hence, for a suitable choice of ν𝜖 (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 Ok×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 > νε, since increasing the order n of \(\mathcal {R}_{n}\), the order of the identity block increases, while A1,1 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 ν𝜖 can be chosen in such a way that also \(\|\mathcal {R}_{n}^{-1}\mathcal {E}_{n}\|<\epsilon \) holds true for any n > ν𝜖. 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 A1,2. Nevertheless, in terms of the column vectors, denoted by \( \mathbf {e}_{i}\in {\mathbb R}^{\nu _{\varepsilon }+1}\) the ith 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 A1,1

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

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,…,ν𝜖, 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 αk,βk,γk,wk, we can conclude that, under a suitable choice of νε, 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 > ν𝜖, 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 > νε 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.

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De Bonis, M.C., Occorsio, D. & Themistoclakis, W. Filtered interpolation for solving Prandtl’s integro-differential equations. Numer Algor 88, 679–709 (2021). https://doi.org/10.1007/s11075-020-01053-x

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Keywords

  • Prandtl equation
  • Hypersingular integral equations
  • Polynomial interpolation
  • Filtered approximation
  • De la Vallée Poussin mean
  • Hölder-Zygmund spaces
  • Chebyshev nodes

Mathematics Subject Classification (2010)

  • 41A10
  • 65D05
  • 33C45