For the central convergence result of this section, Theorem 3.1, we work with discrete tangent-point energies \(\tilde{\mathcal {E}^{n}_{q}}\) with the larger effective domain \(\tilde{\mathcal {B}}_{n}\) (see Definition 2.7(i)), instead of with \(\mathcal {E}^{n}_{q}\) introduced in (8) of the introduction, whose effective domain \(\mathcal {B}_{n}\) is defined by the constraint (7). In other words,
$$\begin{aligned} \tilde{\mathcal {E}}_{q}^{n}(\gamma ):=\textstyle\begin{cases} \sum_{i=0}^{n-1} \sum_{j=0,j\neq i}^{n-1} \left(\frac{2 \operatorname{dist}(l(q_{j}),q_{i})}{ \vert q_{i}-q_{j} \vert ^{2}} \right)^{q} \lambda _{i} \lambda _{j} & \text{if }\gamma \in \tilde{\mathcal{B}}_{n},\\ \infty & \text{otherwise.} \end{cases}\displaystyle \end{aligned}$$
(18)
These discrete energies evaluated on a sequence \((\beta _{n} )_{n \in \mathbb{N}}\) of proper γ-interpolating and balanced biarc curves converge with a certain rate to the continuous \(\mathrm {TP}_{q}\)-energy of γ if γ is sufficiently smooth. Some of the ideas in the proof of the theorem are based on [26, Proposition 3.1] by Scholtes.
Theorem 3.1
Let \(c_{1},c_{2}>0\) and \(\gamma \in C^{1,1}_{\mathrm{ {ia}}} ( \mathbb{R}/ L\mathbb{Z},\mathbb{R}^{3} )\), and \((\mathcal{M}_{n} )_{n \in \mathbb{N}}\) with \(\mathcal{M}_{n}= \{s_{n,0},\dots, s_{n,n} \}\) be a \((c_{1}-c_{2})\)-distributed sequence of partitions of \(\mathbb{R}/ L\mathbb{Z}\) (see Definition 2.8). Then, there is a constant \(C>0\) depending on \(q,c_{1},c_{2}\), and γ, such that for a sequence \((\beta _{n} )_{n \in \mathbb{N}}\) of proper γ-interpolating and balanced biarc curves interpolating the point-tangent data
$$\begin{aligned} \bigl( \bigl( \bigl[\gamma (s_{n,i} ),\gamma ' (s_{n,i} ) \bigr], \bigl[\gamma (s_{n,i+1} ),\gamma ' (s_{n,i+1} ) \bigr] \bigr) \bigr)_{i=0, \dots, n-1} \end{aligned}$$
with \(\beta _{n} \in \tilde{\mathcal{B}}_{n}\) for all \(n \in \mathbb{N}\), there is an index \(N \in \mathbb{N}\) with
$$\begin{aligned} \bigl\vert \mathrm {TP}_{q} (\gamma ) -\tilde{\mathcal {E}}_{q}^{n} (\beta _{n} ) \bigr\vert \leq \frac{C \ln (n)}{n} \quad \textit{for any }n\ge N. \end{aligned}$$
Note that in particular, the convergence rate \(\frac{\ln (n)}{n}\) implies the convergence rate \(\frac{1}{n^{1-\varepsilon}}\) for any given \(\varepsilon >0\).
Proof of Theorem 3.1
Set \(\Upsilon =4\frac{c_{2}}{c_{1}}\) and define for \(i,j \in \{0, \dots, n\}\) the periodic index distance
$$\begin{aligned} \vert i-j \vert _{n}:=\min \bigl\{ \vert i-j \vert ,n- \vert i-j \vert \bigr\} . \end{aligned}$$
We then decompose
$$\begin{aligned} \begin{aligned} \mathrm {TP}_{q} (\gamma )-\tilde{ \mathcal {E}}_{q}^{n} (\beta _{n} ) ={}& \sum _{i=0}^{n-1}\sum_{j, \vert i-j \vert _{n}\leq \Upsilon} \int _{s_{n,j}}^{s_{n,j+1}} \int _{s_{n,i}}^{s_{n,i+1}} \biggl(2\frac{\operatorname{dist}(l (\gamma (t ) ),\gamma (s ) )}{ \vert \gamma (s )-\gamma (t ) \vert ^{2}} \biggr)^{q} \,\mathrm {d}s \,\mathrm {d}t \\ &{}-\sum_{i=0}^{n-1}\sum _{j, 0< \vert i-j \vert _{n}\leq \Upsilon} \biggl(2\frac{\operatorname{dist}(l (\gamma (s_{n,j} ) ),\gamma (s_{n,i} ) )}{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2}} \biggr)^{q} \lambda _{n,i}\lambda _{n,j} \\ &{}+2^{q} \sum_{i=0}^{n-1}\sum _{j, \vert i-j \vert _{n}> \Upsilon} (A_{i,j}+B_{i,j}+C_{i,j} ), \end{aligned} \end{aligned}$$
(19)
with
$$\begin{aligned} &A_{i,j}:= \int _{s_{n,j}}^{s_{n,j+1}} \int _{ s_{n,i}}^{s_{n,i+1}} \frac{\operatorname{dist}(l (\gamma (t ) ),\gamma (s ) )^{q}-\operatorname{dist}(l (\gamma (s_{n,j} ) ),\gamma (s_{n,i} ) )^{q}}{ \vert \gamma (s )-\gamma (t ) \vert ^{2q}} \,\mathrm {d}s \, \mathrm {d}t, \\ &B_{i,j}:= \int _{s_{n,j}}^{s_{n,j+1}} \int _{ s_{n,i}}^{s_{n,i+1}} \biggl( \frac{\operatorname{dist}(l (\gamma (s_{n,j} ) ),\gamma (s_{n,i} ) )^{q}}{ \vert \gamma (s )-\gamma (t ) \vert ^{2q}}- \frac{\operatorname{dist}(l (\gamma (s_{n,j} ) ),\gamma (s_{n,i} ) )^{q}}{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2q}} \biggr) \,\mathrm {d}s \,\mathrm {d}t, \\ &C_{i,j}:= \frac{\operatorname{dist}(l (\gamma (s_{n,j} ) ),\gamma (s_{n,i} ) )^{q}}{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2q}} \bigl[ \vert s_{n,i+1}-s_{n,i} \vert \vert s_{n,j+1}-s_{n,j} \vert - \lambda _{n,i}\lambda _{n,j} \bigr]. \end{aligned}$$
Step 1: Since γ is an injective \(C^{1}\)-curve it is bi-Lipschitz (see Lemma B.1), i.e., there exists a constant \(c_{\gamma }\in (0,\infty )\) such that
$$\begin{aligned} \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}}\leq c_{\gamma } \bigl\vert \gamma (t)-\gamma (s) \bigr\vert \quad \text{for any }t, s \in \mathbb{R}. \end{aligned}$$
(20)
Step 2: Now, we give an upper bound for \(2 \frac{\operatorname{dist}(l (\gamma (t ) ),\gamma (s ) )}{ \vert \gamma (s )-\gamma (t ) \vert ^{2}}\) for all \(s,t \in \mathbb{R}\) with \(s\neq t\). Without loss of generality we assume \(t< s\). Then, there exists a number \(k=k(s,t) \in \mathbb{Z}\) satisfying \(|t-s|_{ \mathbb{R}/ L\mathbb{Z}}=|kL+t-s|\). We use the periodicity of γ and \(K:=\|\gamma ''\|_{L^{\infty}}<\infty \) (since \(\gamma \in C^{1,1}(\mathbb{R}/L\mathbb{Z},\mathbb{R}^{3})\simeq W^{1,\infty}( \mathbb{R}/L\mathbb{Z},\mathbb{R}^{3})\)) to estimate
$$\begin{aligned} & \operatorname{dist}\bigl(l \bigl(\gamma (t ) \bigr),\gamma (s ) \bigr) \\ &\quad=\inf _{\mu \in \mathbb{R}} \bigl\vert \gamma (s)-\gamma (t)-\mu \gamma '(t) \bigr\vert \\ &\quad\le \bigl\vert \gamma (s)-\gamma (kL+t)- \bigl(s-(kL+t) \bigr)\gamma '(kL+t) \bigr\vert \\ &\quad\le \int ^{s}_{kL+t} \int ^{u}_{kL+t} \bigl\vert \gamma ''(v) \bigr\vert \,\mathrm {d}v \,\mathrm {d}u \le K (kL+t-s)^{2}=K \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2}, \end{aligned}$$
(21)
where we assumed, without loss of generality, that \(kL+t< s\) for the integrals. Therefore, by means of (20)
$$\begin{aligned} \biggl(2\frac{\operatorname{dist}(l (\gamma (t ) ),\gamma (s ) )}{ \vert \gamma (s )-\gamma (t ) \vert ^{2}} \biggr)^{q} &\leq \bigl(2 c_{\gamma}^{2} K \bigr)^{q} \quad \text{for any }s,t\in \mathbb{R}, s\neq t. \end{aligned}$$
(22)
Define \(C_{1}:= (2 c_{\gamma}^{2} K )^{q} c_{2} (2\Upsilon +1)L\). Applying the calculations above we can estimate the first term on the right-hand side of (19) from above by
$$\begin{aligned} \stackrel {\text{(22)}}{\leq }{}& \bigl(2 c_{\gamma}^{2} K \bigr)^{q} \sum _{i=0}^{n-1}\sum_{j, \vert i-j \vert _{n}\leq \Upsilon} \vert s_{n,i+1}-s_{n,i} \vert \vert s_{n,j+1}-s_{n,j} \vert \\ \stackrel {\text{(17)}}{\leq }{}& \bigl(2 c_{\gamma}^{2} K \bigr)^{q} \frac{c_{2}}{n} \sum_{i=0}^{n-1} \vert s_{n,i+1}-s_{n,i} \vert \underbrace{ \sum _{j, \vert i-j \vert _{n}\leq \Upsilon} 1}_{\leq 2\Upsilon +1} \\ \stackrel {}{\leq }{}& \bigl(2 c_{\gamma}^{2} K \bigr)^{q} { \frac{c_{2}}{n}} (2\Upsilon +1) \underbrace{ \sum_{i=0}^{n-1} \vert s_{n,i+1}-s_{n,i} \vert }_{=L} \stackrel {}{=}\frac{C_{1}}{n}. \end{aligned}$$
(23)
Step 3: By Lemma 2.10 there exists a constant \(c_{K}\) only depending on K such that
$$\begin{aligned} \biggl\vert \frac{\lambda _{n,i}}{ \vert s_{n,i+1}-s_{n,i} \vert }-1 \biggr\vert \leq c_{K} \vert s_{n,i+1}-s_{n,i} \vert ^{2} \end{aligned}$$
for all \(n \geq N\) and \(i=0,\dots,n-1\), where N depends on the given sequence of biarc curves. Using the fact that \(\vert s_{n,i+1}-s_{n,i} \vert \leq \frac{L}{2}\) yields
$$\begin{aligned} \lambda _{n,i} \leq \underbrace{c_{K} \biggl( \frac{L}{2} \biggr)^{2}}_{=:d_{K}} \vert s_{n,i+1}-s_{n,i} \vert \quad \text{for }n\geq N \text{ and } i=0, \dots, n-1. \end{aligned}$$
(24)
Without loss of generalization we can assume that \(d_{K} \geq 1\). Define \(C_{2}:=d_{K}^{2} C_{1}\). Thus,
$$\begin{aligned} & \sum_{i=0}^{n-1}\sum _{j, 0< \vert i-j \vert _{n}\leq \Upsilon} \biggl(2\frac{\operatorname{dist}(l (\gamma (s_{n,j} ) ),\gamma (s_{n,i} ) )}{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2}} \biggr)^{q} \lambda _{n,i}\lambda _{n,j} \\ &\quad \stackrel {\text{(24)}}{\leq } d_{K}^{2} \sum _{i=0}^{n-1}\sum_{j, 0< \vert i-j \vert _{n}\leq \Upsilon} \biggl(2\frac{\operatorname{dist}(l (\gamma (s_{n,j} ) ),\gamma (s_{n,i} ) )}{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2}} \biggr)^{q} \vert s_{n,i+1}-s_{n,i} \vert \vert s_{n,j+1}-s_{n,j} \vert \\ &\quad \stackrel {\text{(22)}}{\leq } { d_{K}^{2} \bigl(2c_{\gamma}^{2}K \bigr)^{q} \sum _{i=0}^{n-1}\sum_{j, 0< \vert i-j \vert _{n}\leq \Upsilon} \vert s_{n,i+1}-s_{n,i} \vert \vert s_{n,j+1}-s_{n,j} \vert } \overset{\text{(23)}}{\le} \frac{C_{2}}{n}, \end{aligned}$$
(25)
which deals with the second term on the right-hand side of (19).
Step 4: We assume from now on that \(|i-j|_{n}>\Upsilon \). The sequence \((\mathcal{M}_{n})_{n}\) is assumed to be \((c_{1}-c_{2})\)-distributed, so that in view of (17)
$$\begin{aligned} \vert s_{n,k+1}-s_{n,k} \vert = \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}}\quad \text{for any }n\in \mathbb{N}\ \text{and}\ k=0, \dots, n-1. \end{aligned}$$
For \(s \in [s_{n,i},s_{n,i+1})\) and \(t \in [s_{n,j},s_{n,j+1})\) with \(i\neq j\) we use \(|s-t|_{\mathbb{R}/L\mathbb{Z}}\le |s-s_{n,i}|_{ \mathbb{R}/ L\mathbb{Z}} + |s_{n,i}- s_{n,j}|_{ \mathbb{R}/ L\mathbb{Z}}+|s_{n,j}-t|_{ \mathbb{R}/ L\mathbb{Z}}\) to infer the inequality
$$\begin{aligned} \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}} & \stackrel {}{\leq }\vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}+2\max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}} \\ & \stackrel {\text{(17)}}{\leq } \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} + 2\frac{c_{2}}{c_{1}} \underbrace{ \min _{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}}}_{\leq \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}} \\ & \stackrel {}{\leq }\biggl(1+2\frac{c_{2}}{c_{1}} \biggr) \vert s_{n,i}- s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}. \end{aligned}$$
From \(|i-j|_{n}>\Upsilon =4\frac{c_{2}}{c_{1}}\) we have in particular
$$\begin{aligned} 2\frac{c_{2}}{c_{1}}< \frac{1}{2} \vert i-j \vert _{n}. \end{aligned}$$
(26)
Then, similarly as before,
$$\begin{aligned} \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}} &\stackrel {}{\geq }\vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} - 2\max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}} \\ &\stackrel {\text{(17)}}{\geq } \vert i-j \vert _{n} \min_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}}-{ \frac{2c_{2}}{c_{1}}}\min_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}} \\ & \stackrel {}{=}\biggl( \vert i-j \vert _{n}-\frac{2c_{2}}{c_{1}} \biggr) \min_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}} \\ & \overset{\text{(26)}}{>} \frac{1}{2} \vert i-j \vert _{n}\min_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}} \stackrel {}{\geq }\frac{c_{1}}{2c_{2}} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}. \end{aligned}$$
In total, we conclude for \(|i-j|_{n}>\Upsilon \)
$$\begin{aligned} \frac{c_{1}}{2c_{2}} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} \leq \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}} \leq \biggl(1+2 \frac{c_{2}}{c_{1}} \biggr) \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} \end{aligned}$$
(27)
for \(s\in [s_{n,i},s_{n,i+1})\) and \(t\in [s_{n,j},s_{n,j+1})\), which we consider also in Steps 5 and 6.
Step 5: In order to estimate \(A_{i,j}\), we initially estimate for arbitrary \(a,b\geq 0\)
$$\begin{aligned} \bigl\vert b^{q}-a^{q} \bigr\vert = \biggl\vert \int _{a}^{b} \frac{d}{dx}x^{q} \, \mathrm {d}x \biggr\vert = \biggl\vert \int _{a}^{b} qx^{q-1} \,\mathrm {d}x \biggr\vert \leq q \vert b-a \vert \max \{a,b\}^{q-1}, \end{aligned}$$
(28)
since the function \(f: [0,\infty ) \to [0,\infty ), x \to x^{q-1}\) is nondecreasing for \(q\geq 2\). We abbreviate \(d(\cdot,\cdot ):=\operatorname{dist}(l(\gamma (\cdot )),\gamma (\cdot ))\) and use estimate (28) to find for \(s \in [s_{n,i},s_{n,i+1})\) and \(t \in [s_{n,j},s_{n,j+1})\)
$$\begin{aligned} \bigl\vert d^{q}(t,s)-d^{q}(s_{n,j}, s_{n,i}) \bigr\vert \leq q \bigl\vert d(t,s)-d(s_{n,j},s_{n,i}) \bigr\vert \bigl(\max \bigl\{ d(t,s),d(s_{n,j},s_{n,i}) \bigr\} \bigr)^{q-1}. \end{aligned}$$
(29)
Furthermore, combining (21) with (27) yields
$$\begin{aligned} \begin{aligned} & d(t,s)\overset{\text{ (21)}}{\leq} K \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2} \stackrel {\text{ (27)}}{\leq } K \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{2} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2}, \\ &d(s_{n,j},s_{n,i})\leq K \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2}\leq K \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{2} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2}. \end{aligned} \end{aligned}$$
(30)
Hence,
$$\begin{aligned} \bigl(\max \bigl\{ d(t,s),d(s_{n,j},s_{n,i}) \bigr\} \bigr)^{q-1} \leq K^{q-1} \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{2q-2} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2q-2}. \end{aligned}$$
(31)
Moreover, we estimate again by virtue of (27) now for \(s:=s_{n,i}\)
$$\begin{aligned} \vert t-s_{n,i} \vert _{ \mathbb{R}/ L\mathbb{Z}}\stackrel {\text{(27)}}{\leq } \biggl(1+2\frac{c_{2}}{c_{1}} \biggr) \vert s_{n,i}- s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}, \end{aligned}$$
and we use (17) to find for \(t\in [s_{n,j},s_{n,j+1})\)
$$\begin{aligned} \vert t-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} &\stackrel {}{\leq }\max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}} \\ &\stackrel {\text{(17)}}{\leq }{ \frac{c_{2}}{c_{1}}} \min _{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert _{ \mathbb{R}/ L\mathbb{Z}}\stackrel {}{\leq }\biggl(1+2\frac{c_{2}}{c_{1}} \biggr) \vert s_{n,i}- s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}. \end{aligned}$$
Combining these last two estimates with (27) leads to
$$\begin{aligned} \begin{aligned} &\max \bigl\{ \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}}, \vert t-s_{n,i} \vert _{ \mathbb{R}/ L\mathbb{Z}} \bigr\} + \vert t-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}+ \vert t-s_{n,i} \vert _{ \mathbb{R}/ L\mathbb{Z}} \\ &\quad\leq 3 \biggl(1+2\frac{c_{2}}{c_{1}} \biggr) \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} \quad \text{for }s\in [s_{n,i},s_{n,i+1}), t\in [s_{n,j}, s_{n,j+1}). \end{aligned} \end{aligned}$$
(32)
For arbitrary \(\tau \in \mathbb{R}\) the mapping \(P_{\gamma '(\tau )}: \mathbb{R}^{3}\to \mathbb{R}\gamma '(\tau )\) defined as
$$\begin{aligned} P_{\gamma '(\tau )} (v):= \bigl\langle v, \gamma '(\tau ) \bigr\rangle \gamma '(\tau ),\quad \text{for }v\in \mathbb{R}^{3} \end{aligned}$$
(33)
is the orthogonal projection onto the subspace \(\mathbb{R}\gamma '(\tau )\) since \(|\gamma '|=1\), and we have
$$\begin{aligned} \bigl\vert P_{\gamma '(\tau )} (v) -v \bigr\vert &\leq \vert w-v \vert \quad \text{for all } w \in \mathbb{R}\gamma '(\tau ), v\in \mathbb{R}^{3}. \end{aligned}$$
(34)
Moreover, we have for any \(\tau,\sigma \in \mathbb{R}\)
$$\begin{aligned} d(\tau,\sigma )=\operatorname{dist}\bigl(l \bigl(\gamma (\tau ) \bigr),\gamma (\sigma ) \bigr) &= \bigl\vert P_{\gamma '(\tau )} \bigl(\gamma (\sigma )-\gamma (\tau ) \bigr) - \bigl(\gamma (\sigma )-\gamma (\tau ) \bigr) \bigr\vert . \end{aligned}$$
(35)
Furthermore, we calculate for \(s\in [s_{n,i},s_{n,i+1})\) and \(t\in [s_{n,j},s_{n,j+1})\) using the linearity of the projection
$$\begin{aligned} & P_{\gamma '(t)} \bigl(\gamma (s) -\gamma (t) \bigr)-P_{\gamma ' (s_{n,j} )} \bigl( \gamma (s_{n,i} )-\gamma (s_{n,j} ) \bigr) \\ &\quad \stackrel {}{=}P_{\gamma '(t)} \bigl(\gamma (s)-\gamma (s_{n,i}) \bigr) +P_{\gamma '(t)} \bigl(\gamma (s_{n,i})-\gamma (t) \bigr) \\ &\qquad{}-P_{\gamma ' (s_{n,j} )} \bigl(\gamma (t)-\gamma (s_{n,j}) \bigr) -P_{\gamma ' (s_{n,j} )} \bigl(\gamma (s_{n,i})-\gamma (t) \bigr) \\ &\quad \stackrel {\text{(33)}}{=} P_{\gamma '(t)} \bigl(\gamma (s)-\gamma (s_{n,i} ) \bigr) - P_{\gamma ' (s_{n,j} )} \bigl( \gamma (t)-\gamma (s_{n,j} ) \bigr) \\ &\qquad{}+ \bigl\langle \gamma (s_{n,i} ) - \gamma (t),\gamma '(t)-\gamma ' (s_{n,j} ) \bigr\rangle \gamma '(t) \\ &\qquad{}+ \bigl\langle \gamma (s_{n,i} ) - \gamma (t), \gamma ' (s_{n,j} ) \bigr\rangle \bigl( \gamma '(t) - \gamma ' (s_{n,j} ) \bigr) . \end{aligned}$$
(36)
In conclusion, by (35) and the elementary inequality \(\Vert a|-|b\Vert \le |a-b|\), this yields for the expression \(|d(t,s)-d(s_{n,j},s_{n,i})|\) (for \(s\in [s_{n,i},s_{n,i+1})\) and \(t\in [s_{n,j},s_{n,j+1})\)) the upper bound
$$\begin{aligned} \bigl\vert P_{\gamma '(t)} \bigl(\gamma (s) - \gamma (t) \bigr)-P_{ \gamma ' (s_{n,j} )} \bigl(\gamma (s_{n,i} ) - \gamma (s_{n,j} ) \bigr)- \bigl(\gamma (s) - \gamma (s_{n,i} ) \bigr)+ \bigl(\gamma (t) - \gamma (s_{n,j} ) \bigr) \bigr\vert , \end{aligned}$$
which in turn by means of (36) and (34) can be bounded from above by
$$\begin{aligned} & \bigl\vert P_{\gamma '(t)} \bigl(\gamma (s) - \gamma (s_{n,i} ) \bigr) - \bigl(\gamma (s) - \gamma (s_{n,i} ) \bigr) \bigr\vert \\ &\qquad{}+ \bigl\vert P_{\gamma ' (s_{n,j} )} \bigl( \gamma (t) - \gamma (s_{n,j} ) \bigr) - \bigl(\gamma (t) - \gamma (s_{n,j} ) \bigr) \bigr\vert \\ &\qquad{}+ \bigl\vert \bigl\langle \gamma (s_{n,i}) - \gamma (t),\gamma '(t) - \gamma ' (s_{n,j} ) \bigr\rangle \gamma '(t) \bigr\vert \\ &\qquad{}+ \bigl\vert \bigl\langle \gamma (s_{n,i} ) - \gamma (t), \gamma ' (s_{n,j} ) \bigr\rangle \bigl(\gamma '(t) - \gamma ' (s_{n,j} ) \bigr) \bigr\vert \\ &\quad \stackrel {\text{(34)}}{\leq } \bigl\vert \bigl(\gamma (s)- \gamma (s_{n,i} ) \bigr)-(s-s_{n,i})\gamma '(t) \bigr\vert + \bigl\vert \bigl(\gamma (t)- \gamma (s_{n,j} ) \bigr)-(t- s_{n,j})\gamma ' (s_{n,j} ) \bigr\vert \\ &\qquad{}+2 \bigl\vert \gamma (s_{n,i} )-\gamma (t) \bigr\vert \bigl\vert \gamma ' (s_{n,j} )-\gamma '(t) \bigr\vert . \end{aligned}$$
The last summand is bounded by \(2K|t-s_{n,i}|_{ \mathbb{R}/ L\mathbb{Z}}|t-s_{n,j}|_{ \mathbb{R}/ L\mathbb{Z}}\) since \(K=\|\gamma ''\|_{L^{\infty}}\) and 1 are the Lipschitz constants of \(\gamma '\) and γ, respectively. The first summand on the right-hand side of the above equals \(|\int _{s_{n,i}}^{s}\int _{t}^{u} \gamma ''(v)\,\mathrm {d}v \,\mathrm {d}u |\), whereas the second is bounded by \(\int _{s_{n,j}}^{t}\int _{s_{n,j}}^{u} |\gamma ''(v)| \,\mathrm{d} v \,\mathrm {d}u\), so that we can summarize the estimate
$$\begin{aligned} & \bigl\vert d(t,s)-d(s_{n,j},s_{n,i}) \bigr\vert \\ &\quad \stackrel {}{\leq }K \vert s_{n,i}-s \vert _{ \mathbb{R}/ L\mathbb{Z}} \max \bigl\{ \vert s-t \vert _{ \mathbb{R}/ L\mathbb{Z}}, \vert t-s_{n,i} \vert _{ \mathbb{R}/ L\mathbb{Z}} \bigr\} \\ &\qquad{}+ K \vert t-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2} +2K \vert t- s_{n,i} \vert _{ \mathbb{R}/ L\mathbb{Z}} \vert t-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} \\ &\quad \stackrel {}{\leq }2K \max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \\ &\qquad{}\times \bigl[\max \bigl\{ \vert s-t \vert _{ \mathbb{R}/ L\mathbb{Z}}, \vert t- s_{n,i} \vert _{ \mathbb{R}/ L\mathbb{Z}} \bigr\} + \vert t-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} + \vert t-s_{n,i} \vert _{ \mathbb{R}/ L\mathbb{Z}} \bigr] \\ &\quad \stackrel {\text{(32)}}{\leq } 6 K \biggl(1+2\frac{c_{2}}{c_{1}} \biggr) \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} \max _{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert . \end{aligned}$$
(37)
Inserting (31) and (37) into (29) yields
$$\begin{aligned} & \bigl\vert d^{q}(t,s)-d^{q}(s_{n,j}, s_{n,i}) \bigr\vert \\ &\quad\leq { 6qK^{q} \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{2q-1}} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2q-1} \max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert . \end{aligned}$$
(38)
In order to obtain an estimate for the denominator of \(A_{i,j}\) we consider
$$\begin{aligned} \bigl\vert \gamma (s)-\gamma (t) \bigr\vert ^{2q} \stackrel {\text{(20)}}{\geq } \frac{1}{c_{\gamma}^{2q}} \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2q} \stackrel {\text{(27)}}{\geq } \biggl(\frac{c_{1}}{2c_{2} c_{\gamma}} \biggr)^{2q} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2q}. \end{aligned}$$
(39)
Setting \(C_{A}:= \frac{c_{2}^{3}}{c_{1}} 6 q K^{q} (1+2\frac{c_{2}}{c_{1}})^{2q-1} (\frac{2c_{2} c_{\gamma}}{c_{1}})^{2q}\) we obtain from (38) and (39)
$$\begin{aligned} \vert A_{i,j} \vert & \stackrel {\text{ (38)},\text{(39)}}{\leq } \frac{c_{1}}{ (c_{2} )^{3}} C_{A} \int _{s_{n,j}}^{s_{n,j+1}} \int _{s_{n,i}}^{s_{n,i+1}}\frac{ \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2q-1}\max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert }{ \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2q}}\,\mathrm {d}s \,\mathrm {d}t \\ &\stackrel {}{\leq }\frac{c_{1}}{ (c_{2} )^{3}} C_{A} \Bigl(\max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \Bigr)^{3} \frac{1}{ \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}} \\ &\stackrel {\text{(17)}}{\leq } c_{1} C_{A} \frac{1}{n^{3}}\frac{1}{ \vert i-j \vert _{n}\min_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert } \stackrel {\text{(17)}}{\leq } C_{A} \frac{1}{n^{2}} \frac{1}{ \vert i-j \vert _{n}}. \end{aligned}$$
(40)
Step 6: To estimate \(B_{i,j}\), we use (28) and twice (27) leading to
$$\begin{aligned} & \bigl\vert \bigl|\gamma (s_{n,i} )-\gamma (s_{n,j} ) \bigr\vert ^{2q}- \bigl\vert \gamma (s )-\gamma (t ) \bigr\vert ^{2q}\bigr| \\ &\quad \stackrel {}{=}\bigl\vert \bigl\vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \bigr\vert ^{q}+ \bigl\vert \gamma (s )-\gamma (t ) \bigr\vert ^{q} \bigr\vert \\ &\qquad{}\times\bigl| \bigl\vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \bigr\vert ^{q}- \vert \gamma (s )-\gamma (t ) \vert ^{q}\bigr| \\ &\quad \stackrel {\text{(28)}}{\leq } q \bigl( \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{q}+ \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}}^{q} \bigr)\bigl|\bigl|\gamma (s_{n,i} )-\gamma (s_{n,j} )\bigr|-\bigl|\gamma (s )-\gamma (t ) \bigr\vert \bigr\vert \\ &\qquad{}\times \max \bigl\{ \bigl\vert \gamma (s )-\gamma (t ) \bigr\vert , \bigl\vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \bigr\vert \bigr\} ^{q-1} \\ &\quad \stackrel {\text{(27)}}{\leq } 2q \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{q}|s_{n,i}-s_{n,j}|_{ \mathbb{R}/ L\mathbb{Z}}^{q} \bigl\vert \gamma (s_{n,i} )-\gamma (s)+\gamma (t)-\gamma (s_{n,j} ) \bigr\vert \\ &\qquad{}\times \max \bigl\{ \vert t-s \vert _{ \mathbb{R}/ L\mathbb{Z}}, \vert s_{n,i}- s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} \bigr\} ^{q-1} \\ &\quad \stackrel {\text{(27)}}{\leq } 2q \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{2q-1}|s_{n,i}-s_{n,j}|_{ \mathbb{R}/ L\mathbb{Z}}^{2q-1} \bigl( \bigl\vert \gamma (s )-\gamma (s_{n,i} ) \bigr\vert + \bigl\vert \gamma (t )-\gamma (s_{n,j} ) \bigr\vert \bigr) \\ &\quad \stackrel {}{\leq }2q \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{2q-1} |s_{n,i}- s_{n,j}|_{ \mathbb{R}/ L\mathbb{Z}}^{2q-1} \bigl( \vert s-s_{n,i} \vert _{ \mathbb{R}/ L\mathbb{Z}}+ \vert t-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}} \bigr) \\ &\quad \stackrel {}{\leq }4q \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{2q-1}|s_{n,i}- s_{n,j}|_{ \mathbb{R}/ L\mathbb{Z}}^{2q-1} \max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert . \end{aligned}$$
(41)
Thus, by (30), (41), and (39),
$$\begin{aligned} \vert B_{i,j} \vert \stackrel {\text{(30)}}{\leq }{}& K^{q} \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{2q} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{2q} \\ &{}\times \int _{s_{n,j}}^{s_{n,j+1}} \int _{s_{n,i}}^{s_{n,i+1}} \biggl\vert \frac{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2q}- \vert \gamma (s )-\gamma (t ) \vert ^{2q}}{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2q} \vert \gamma (s )- \gamma (t ) \vert ^{2q}} \biggr\vert \,\mathrm {d}s \,\mathrm {d}t \\ \stackrel {\text{(41)}}{\leq }{}& 4q K^{q} \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{4q-1} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{4q-1}\max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \\ &{} \times \int _{s_{n,j}}^{s_{n,j+1}} \int _{s_{n,i}}^{s_{n,i+1}} \frac{1}{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2q} \vert \gamma (s )-\gamma (t ) \vert ^{2q}} \,\mathrm {d}s \, \mathrm {d}t \\ \stackrel {\text{(39)}}{\leq }{}& \biggl(\frac{2c_{2} c_{\gamma}}{c_{1}} \biggr)^{2q}c_{\gamma}^{2q} 4q K^{q} \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{4q-1} \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{4q-1} \max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \\ &{} \times \int _{s_{n,j}}^{s_{n,j+1}} \int _{s_{n,i}}^{s_{n,i+1}} \frac{1}{ \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}^{4q}} \,\mathrm {d}s \, \mathrm {d}t \\ \stackrel {}{\leq }{}& \biggl(\frac{2c_{2} c_{\gamma}}{c_{1}} \biggr)^{2q}c_{\gamma}^{2q} 4q K^{q} \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{4q-1} \Bigl( \max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \Bigr)^{3} \frac{1}{ \vert s_{n,i}-s_{n,j} \vert _{ \mathbb{R}/ L\mathbb{Z}}} \\ \stackrel {\text{(17)}}{\leq }{}& (c_{2} )^{3} \biggl(\frac{2c_{2} c_{\gamma}}{c_{1}} \biggr)^{2q}c_{\gamma}^{2q} 4q K^{q} \biggl(1+2\frac{c_{2}}{c_{1}} \biggr)^{4q-1} \frac{1}{n^{3}} \frac{1}{ \vert i-j \vert _{n}\min_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert } \\ \stackrel {\text{(17)}}{\leq }{}&\frac{C_{B}}{n^{2}} \frac{1}{ \vert i-j \vert _{n}} \end{aligned}$$
(42)
with \(C_{B}:=\frac{(c_{2})^{3}}{c_{1}} ( \frac{2c_{2} c_{\gamma}}{c_{1}} )^{2q}c_{\gamma}^{2q} 4q K^{q} (1+2\frac{c_{2}}{c_{1}} )^{4q-1}\).
Step 7: The expression \(\sum_{k=1}^{n} \frac{1}{k}-\ln (n)\) converges for \(n \to \infty \) to the Euler–Mascheroni constant; see [18, p. xix]. Thus, there exists a constant \(c_{l}\in (0,\infty )\) such that \(\vert \sum_{k=1}^{n} \frac{1}{k}-\ln (n) \vert \leq c_{l}\) for all \(n \in \mathbb{N}\). This leads for \(n \geq 1\) to
$$\begin{aligned} & \sum_{i=0}^{n-1} \sum _{j, \vert i-j \vert _{n}>\Upsilon} \bigl( \vert A_{i,j} \vert + \vert B_{i,j} \vert \bigr) \\ &\quad \stackrel {\text{(40)},\text{ (42)}}{\leq } \frac{2\max \{C_{A},C_{B}\}}{n^{2}}\sum_{i=0}^{n-1} \sum_{j, \vert i-j \vert _{n}>\Upsilon} \frac{1}{ \vert i-j \vert _{n}} \\ &\quad \stackrel {}{\leq }\frac{4\max \{C_{A},C_{B}\}}{n^{2}}\sum_{i=0}^{n-1} \sum_{k=1}^{n} \frac{1}{k} \\ &\quad \stackrel {}{=}\frac{4\max \{C_{A},C_{B}\}}{n} \Biggl(\sum_{k=1}^{n} \frac{1}{k}-\ln (n) \Biggr)+4\max \{C_{A},C_{B}\} \frac{\ln (n)}{n} \stackrel {}{\leq }\frac{C_{AB}\ln (n)}{n} \end{aligned}$$
(43)
with \(C_{AB}:=8 \max \{c_{l},1\}\max \{C_{A},C_{B}\}\).
Step 8: Recall from Step 3 that
$$\begin{aligned} { \bigl\vert \lambda _{n,j}- \vert s_{n,j+1}- s_{n,j} \vert \bigr\vert \leq c_{K} \vert s_{n,j+1}- s_{n,j} \vert ^{3}} \leq c_{K} \Bigl(\max _{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \Bigr)^{3} \end{aligned}$$
(44)
holds for all \(n \geq N\) and \(j=0,\dots, n-1\), where N depends on the sequence \((\beta _{n})_{n \in \mathbb{N}}\). From (22) and (24) we obtain from (44)
$$\begin{aligned} \vert C_{i,j} \vert \stackrel {}{\leq }{} & \frac{\operatorname{dist}(l (\gamma (s_{n,j} ) ),\gamma (s_{n,i} ) )^{q}}{ \vert \gamma (s_{n,i} )-\gamma (s_{n,j} ) \vert ^{2q}} \bigl\vert \vert s_{n,i+1}-s_{n,i} \vert \vert s_{n,j+1}-s_{n,j} \vert -\lambda _{n,i}\lambda _{n,j}\bigr\vert \\ \stackrel {\text{(22)}}{\leq }{}& \bigl(c_{\gamma}^{2} K \bigr)^{q} \bigl[ \vert s_{n,i+1}-s_{n,i} \vert \bigl\vert \vert s_{n,j+1}-s_{n,j} \vert -\lambda _{n,j} \bigr\vert \\ &{}+\lambda _{n,j} \bigl\vert \vert s_{n,i+1}-s_{n,i} \vert -\lambda _{n,i} \bigr\vert \bigr] \\ \stackrel {\text{(24)}}{\leq }{}& \bigl(c_{\gamma}^{2} K \bigr)^{q} d_{K} \bigl[ \vert s_{n,i+1}-s_{n,i} \vert \bigl\vert \vert s_{n,j+1}-s_{n,j} \vert -\lambda _{n,j} \bigr\vert \\ &{}+ \vert s_{n,j+1}-s_{n,j} \vert \bigl\vert \vert s_{n,i+1}-s_{n,i} \vert -\lambda _{n,i} \bigr\vert \bigr] \\ \stackrel {\text{(44)}}{\leq }{}& \bigl(c_{\gamma}^{2} K \bigr)^{q} d_{K} c_{K} \Bigl(\max _{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \Bigr)^{3} \\ &{}\times \bigl[ \vert s_{n,i+1}-s_{n,i} \vert + \vert s_{n,j+1}- s_{n,j} \vert \bigr] \\ \stackrel {}{\leq }{}& 2 \bigl(c_{\gamma}^{2} K \bigr)^{q} d_{K} c_{K} \Bigl(\max_{k=0, \dots, n-1} \vert s_{n,k+1}-s_{n,k} \vert \Bigr)^{4} \stackrel {\text{ (17)}}{\leq }\frac{C_{C}}{n^{4}}, \end{aligned}$$
(45)
with \(C_{C}:=2 (c_{2} )^{4} (c_{\gamma}^{2} K )^{q} d_{K} c_{K}\) for all \(n \geq N\). We then conclude that
$$\begin{aligned} \sum_{i=0}^{n-1} \sum _{j, \vert i-j \vert _{n}>\Upsilon} \vert C_{i,j} \vert \stackrel {\text{ (45)}}{\leq } \frac{C_{C}}{n^{4}}\sum_{i=0}^{n-1} \sum_{j=0}^{n-1} 1 = \frac{C_{C}}{n^{2}}. \end{aligned}$$
(46)
Step 9: Inserting (23), (25), (43), and (46) into (19) yields
$$\begin{aligned} \bigl\vert \mathrm {TP}_{q} (\gamma )-\tilde{\mathcal {E}}_{q}^{n} (\beta _{n} ) \bigr\vert \leq & \frac{C_{1}}{n}+ \frac{C_{2}}{n}+2^{q} \biggl( \frac{C_{AB} \ln (n)}{n} + \frac{C_{C}}{n^{2}} \biggr) \leq \frac{C \ln (n)}{n} \quad\text{for } n \geq N, \end{aligned}$$
with \(C:=4\max \{C_{1},C_{2},2^{q} C_{AB}, 2^{q} C_{C}\}\), which gives the desired result. □