Superconvergence of Jacobi–Gauss-Type Spectral Interpolation

Abstract

In this paper, we extend the study of superconvergence properties of Chebyshev-Gauss-type spectral interpolation in Zhang (SIAM J Numer Anal 50(5):2966–2985, 2012) to general Jacobi–Gauss-type interpolation. We follow the same principle as in Zhang (SIAM J Numer Anal 50(5):2966–2985, 2012) to identify superconvergence points from interpolating analytic functions, but rigorous error analysis turns out much more involved even for the Legendre case. We address the implication of this study to functions with limited regularity, that is, at superconvergence points of interpolating analytic functions, the leading term of the interpolation error vanishes, but there is no gain in order of convergence, which is in distinctive contrast with analytic functions. We provide a general framework for exponential convergence and superconvergence analysis. We also obtain interpolation error bounds for Jacobi–Gauss-type interpolation, and explicitly characterize the dependence of the underlying parameters and constants, whenever possible. Moreover, we provide illustrative numerical examples to show tightness of the bounds.

Appendices

Appendix A: Jacobi Polynomials

The Jacobi polynomials satisfy the derivative relation (see [15, (4.21.7)]):

$$\begin{aligned} \partial _xP_N^{(\alpha ,\beta )}(x)=\frac{1}{2}(N+\alpha +\beta +1)P_{N-1}^{(\alpha +1,\beta +1)}(x), \end{aligned}$$

(6.1)

and there holds (see [15, (4.5.4)])):

$$\begin{aligned} P_{N}^{(\alpha ,\beta +1)}(x)=\frac{2}{2N+\alpha +\beta +2} \frac{(N+\beta +1)P_N^{(\alpha ,\beta )}(x)+(N+1)P_{N+1}^{(\alpha , \beta )}(x)}{1+x}.\qquad \end{aligned}$$

(6.2)

Another recurrence relation reads (see [15, (4.5.5)–(4.5.6)]):

$$\begin{aligned} (1-x^2)\partial _xP_{N}^{(\alpha ,\beta )}(x) =AP_{N-1}^{(\alpha ,\beta )}(x)+BP_{N}^{(\alpha ,\beta )}(x) +CP_{N+1}^{(\alpha ,\beta )}(x), \end{aligned}$$

(6.3)

where

$$\begin{aligned} A&= \frac{2(N+\alpha )(N+\beta )(N+\alpha +\beta +1)}{(2N+\alpha +\beta )(2N +\alpha +\beta +1)},\;\;\;\; B=\frac{2(\alpha -\beta )N(N+\alpha +\beta +1)}{(2N+\alpha +\beta )(2N +\alpha +\beta +2)},\\ C&= -\frac{2N(N+1)(N+\alpha +\beta +1)}{(2N+\alpha +\beta +1)(2N +\alpha +\beta +2)}. \end{aligned}$$

Appendix B: Proof of Theorem 4.2

We first present some necessary lemmas for its proof.

The following formula (see [3, Lemma 12.4.1]) is of paramount importance.

Lemma 7.1

Let $z=(w+w^{-1})/2.$ Then

$$\begin{aligned} P_N(z)=\sum _{k=0}^N g_kg_{N-k}w^{N-2k}, \end{aligned}$$

(7.1)

where

$$\begin{aligned} g_k=\frac{(2k)!}{(k!)^22^{2k}}=\frac{\Gamma (k+1/2)}{\sqrt{\pi }\, k!},\quad k\ge 0. \end{aligned}$$

(7.2)

In fact, the coefficients $\{g_k\}$ relate to the following Laurent series expansion.

Lemma 7.2

We have

$$\begin{aligned} F(w):=(1-w^{-2})^{-1/2}(1+w^{-1})= \sum _{k=0}^\infty \frac{g_k}{w^{2k}}+\sum _{k=0}^\infty \frac{g_k}{w^{2k+1}}, \end{aligned}$$

(7.3)

which converges uniformly for all complex-valued $w$ such that $|w|>1.$

Proof

Recall the binomial expansion:

$$\begin{aligned} (1-w^{-2})^{-1/2}=\sum _{k=0}^\infty \frac{g_k}{w^{2k}},\quad \forall \, |w|>1. \end{aligned}$$

(7.4)

This implies (7.3).$\square$

The key idea of estimating $m_Q$ is to show that for $z\in \mathcal{E }_\rho$ with $|w|=\rho >1,$

$$\begin{aligned} \bigg |F(w)-\frac{Q_{N+1}(z)}{g_{N+1} w^{N+1}}\bigg |\rightarrow 0\;\; \mathrm{as} \;\; N\rightarrow \infty , \end{aligned}$$

(7.5)

and more importantly, we care about the rate it decays. For this purpose, let us split the error term into two parts:

$$\begin{aligned} \bigg |F(w)-\frac{Q_{N+1}(z)}{g_{N+1} w^{N+1}}\bigg |&= \bigg |\bigg (\sum _{k=0}^\infty \frac{g_k}{w^{2k}}- \sum _{k=0}^{N+1} \frac{g_kg_{N+1-k}}{g_{N+1} w^{2k}}\bigg ) \nonumber \\&\quad +\bigg (\sum _{k=0}^\infty \frac{g_k}{ w^{2k+1}}- \sum _{k=0}^{N} \frac{g_kg_{N-k}}{g_{N+1}w^{2k+1}}\bigg ) \bigg |\\&\le R_N^{e}(\rho ) + R_N^{o}(\rho ),\nonumber \end{aligned}$$

(7.6)

where

$$\begin{aligned} R_N^e(\rho )=\sum _{k=0}^{N+1}\frac{|q_k| |g_k|}{\rho ^{2k}}+ \sum _{k=N+2}^\infty \frac{|g_k|}{\rho ^{2k}}; \;\; R_N^o(\rho )=\sum _{k=0}^{N}\frac{|q_{k+1}| |g_k|}{\rho ^{2k+1}}+ \sum _{k=N+1}^\infty \frac{|g_k|}{\rho ^{2k+1}},\qquad \quad \end{aligned}$$

(7.7)

with

$$\begin{aligned} q_k:=q_k(N):=\frac{ g_{N+1-k}}{g_{N+1}}-1,\quad 0\le k\le N+1. \end{aligned}$$

(7.8)

We deduce from (7.2) and (7.8) the following useful properties of $\{g_k\}$ and $\{q_k\}.$

Lemma 7.3

• (i) For $k\ge 0, \, g_k>0,$ and $\{g_k\}$ is strictly decreasing, namely,

  $$\begin{aligned} 1=g_0>g_1>\cdots >g_{k}>g_{k+1}>\cdots . \end{aligned}$$

  (7.9)

• (ii) We have

  $$\begin{aligned} 0=q_0<q_1<\cdots <q_{N}<q_{N+1}. \end{aligned}$$

  (7.10)

  Moreover, $(q_k+1)g_k<1,$ for $1\le k\le N$, and $(q_{k+1}+1)g_k<1$ for $1\le k\le N-1.$ In addition, $(q_{N+1}+1)g_{N+1}=1$, and $q_{N+1}g_N<1$ for $N\ge 2$.

Proof

• (i) It is clear that by (7.2), $g_0=1$ and $0<g_k<1$ for all $k\ge 1.$ Since

  $$\begin{aligned} \frac{g_{k+1}}{g_k}= \frac{k+1/2}{k+1}<1, \end{aligned}$$

  (7.11)

  $\{g_k\}$ is strictly decreasing with respect to $k$.

• (ii) Observe from (7.8) and (i) that $q_0=0$ and $\{q_k\}$ is strictly increasing. A direct calculation leads to that for $1\le k\le N-1,$

  $$\begin{aligned} (q_{k+1}+1)g_k&= \frac{g_{N-k}}{g_{N+1}}g_{k}=\left( \prod _{j=0}^{k-2} \frac{1-\frac{1/2}{k-j}}{1-\frac{1/2}{N+1-j}}\right) \frac{(N-k+2)(N-k+1)}{2(N-k+3/2)(N-k+1/2)}\\&\le \frac{1}{2}\Big (1+\frac{1/2}{N-k+3/2}\Big ) \Big (1+\frac{1/2}{N-k+1/2}\Big )\le \frac{4}{5}<1, \end{aligned}$$

  where we used the fact $1+\frac{1/2}{N-k+3/2}$ and $1+\frac{1/2}{N-k+1/2}$ are strictly increasing with respect to $k$. Note that if $k=1$, the first term in the second identity equals $1$.

By (7.9)–(7.10), we have

$$\begin{aligned} (q_{k+1}+1)g_{k+1}<(q_{k+1}+1)g_k;\quad (q_{k}+1)g_{k}<(q_{k+1}+1)g_k, \end{aligned}$$

which implies $(q_{k}+1)g_k<1$ for $1\le k\le N$.

Next, by (7.2) and (7.8), we have $(q_{N+1}+1)g_{N+1}=1$ and

$$\begin{aligned} q_{N+1}g_{N}=\frac{g_{N}}{g_{N+1}}-g_N=1+\frac{1}{2N+1}-g_N<1, \end{aligned}$$

since $g_N=\frac{\Gamma (N+1/2)}{\sqrt{\pi }N\Gamma (N)} >\frac{1}{\sqrt{\pi }N}>\frac{1}{2N+1}$ for $N\ge 2$. This ends the proof.$\square$

Lemma 7.4

We have

$$\begin{aligned} 0<q_k\le D_{k,N}, \quad 1\le k\le N-1, \end{aligned}$$

(7.12)

where $D_{k,N}$ is defined in (4.17).

Proof

By (7.2) and (7.8), we obtain from (4.2)–(4.3) that

$$\begin{aligned} q_k+1&= \frac{g_{N+1-k}}{g_{N+1}}=\frac{\Gamma (N-k+3/2)}{\Gamma (N-k+2)} \frac{\Gamma (N+2)}{\Gamma (N+3/2)} \\&\le \sqrt{\frac{N+1}{N-k+1}}\, \mathrm{exp}\big (\Upsilon _{N-k+1}^{1/2,1} +\Upsilon _{N+1}^{1,1/2}\big ). \end{aligned}$$

A direct calculation leads to

$$\begin{aligned} \Upsilon _{N-k+1}^{1/2,1}+\Upsilon _{N+1}^{1,1/2}&= \frac{1}{6(2N-2k+1)}+\bigg (\frac{1}{2(2N+1)} +\frac{1}{12(N+1)}\bigg )\\&\le \frac{1}{12(N-k)} + \frac{1}{3N}\le \frac{5}{12 (N-k)},\quad 1\le k\le N-1. \end{aligned}$$

Using the fact $\sqrt{1+x}\le 1+\frac{x}{2}$ (with $x\ge 0$) yields

$$\begin{aligned} \sqrt{\frac{N+1}{N-k+1}}\le 1+\frac{k}{2(N-k+1)}\le 1+\frac{k}{2(N-k)}. \end{aligned}$$

Consequently, we obtain

$$\begin{aligned} q_k+1\le \Big (1+\frac{k}{2(N-k)}\Big )\mathrm{exp}\Big (\frac{5}{12 (N-k)}\Big ), \end{aligned}$$

(7.13)

which gives the desired upper bound.$\square$

Proof of Theorem 4.2

By (7.6),

$$\begin{aligned} \big |Q_{N+1}(z)\big |\ge g_{N+1} \rho ^{N+1}\big \{|F(w)|-\big (R_N^e(\rho )+R_N^o(\rho )\big )\big \}. \end{aligned}$$

(7.14)

As $|w|=\rho ,$ we find

$$\begin{aligned} |F(w)|=|1-w^{-2}|^{-1/2}|1+w^{-1}|\ge \frac{1-\rho ^{-1}}{\sqrt{1+\rho ^{-2}}}=\frac{\rho -1}{\sqrt{1+\rho ^2}}. \end{aligned}$$

(7.15)

By (7.2) and (4.2),

$$\begin{aligned} g_{N+1}\ge \frac{1}{\sqrt{\pi N}} \mathrm{exp}\big (-\Upsilon _N^{2,3/2}\big )\ge \frac{1}{\sqrt{\pi N}} e^{-\frac{5}{6N}}. \end{aligned}$$

(7.16)

We now work on the upper bound of $R_N^e(\rho )+R_N^o(\rho )$ defined in (7.7). Using the properties in Lemma 7.3 and (7.3)–(7.4), we obtain that for some $1<K<N,$

$$\begin{aligned} R_N^e(\rho )&= \sum _{k=1}^{K}\frac{q_k g_k}{\rho ^{2k}}+\sum _{k=K+1}^{N+1}\frac{q_k g_k}{\rho ^{2k}}+ \sum _{k=N+2}^\infty \frac{g_k}{\rho ^{2k}}\\&\le q_K \sum _{k=1}^{K}\frac{g_k}{\rho ^{2k}}+ \sum _{k=K+1}^{N+1}\frac{1}{\rho ^{2k}}+ \sum _{k=N+2}^\infty \frac{1}{\rho ^{2k}}\\&\le q_K \Big ((1-\rho ^{-2})^{-1/2}-1\Big )+ \frac{1}{\rho ^{2K}(\rho ^2-1)}, \end{aligned}$$

and similarly,

$$\begin{aligned} R_N^o(\rho )&= \sum _{k=0}^{K-1}\frac{q_{k+1} g_k}{\rho ^{2k+1}}+\sum _{k=K}^{N}\frac{q_{k+1} g_k}{\rho ^{2k+1}}+ \sum _{k=N+1}^\infty \frac{g_k}{\rho ^{2k+1}}\\&\le q_K \sum _{k=0}^{K-1}\frac{g_k}{\rho ^{2k+1}}+ \sum _{k=K}^{N}\frac{1}{\rho ^{2k+1}}+ \sum _{k=N+1}^\infty \frac{1}{\rho ^{2k+1}}\\&\le q_K \rho ^{-1}(1-\rho ^{-2})^{-1/2}+ \frac{1}{\rho ^{2K-1}(\rho ^2-1)}. \end{aligned}$$

Collecting the terms leads to the upper bound

$$\begin{aligned} R_N^e(\rho )+R_N^o(\rho ) \le \bigg (\sqrt{\frac{\rho +1}{\rho -1}}-1 \bigg ) q_K + \frac{1}{\rho ^{2K}(\rho -1)}. \end{aligned}$$

(7.17)

Thus, a combination of (7.14)–(7.17) yields (4.15).

It remains to show the asymptotic estimate (4.18). Observe that for $N\gg 1,$ if we choose $K=[N^\varepsilon ]$ with $0<\varepsilon <1,$ then

$$\begin{aligned} D_{K,N}=\frac{N^{\varepsilon -1}}{2}+O(N^{-1}). \end{aligned}$$

(7.18)

Therefore,

$$\begin{aligned} A(K,N;\rho )=\frac{\rho -1}{\sqrt{1+\rho ^2}}-\bigg (\sqrt{\frac{\rho +1}{\rho -1}}-1 \bigg )\Big (\frac{N^{\varepsilon -1}}{2}+O(N^{-1})\Big ) -O\big (\rho ^{-(2N^{\varepsilon }+1)}\big ).\nonumber \\ \end{aligned}$$

(7.19)

Thus, for $N\gg 1,$

$$\begin{aligned} A(K,N;\rho )\approx \frac{\rho -1}{\sqrt{1+\rho ^2}}. \end{aligned}$$

Hence, the conclusion follows from (4.15). $\square$