First, we have the following estimate:
$$\begin{aligned} \widetilde{f}_{\mathfrak {M},\mathcal {N}}(\mathbf {x})-f(\mathbf {x}) ={}& \sum_{\mathfrak {n}=1}^{\mathcal {N}} \frac{ \frac{\pi}{ \mathfrak {M}}\sum_{\mathbf {k}=1}^{\mathfrak {M}}\sigma_{\mathbf {k}}\varepsilon _{\mathbf {k}}\phi_{\mathfrak {n}}(\mathbf {x}_{\mathbf {k}})+\overline{\mathcal{G}}_{\mathfrak {n},\mathfrak {M}} }{\int _{0}^{\mathcal {T}_{0}} \mathbf{S}_{\mathfrak {n}}(\mathcal {T}_{0}-\mathfrak {s},\gamma) \mathcal{Q}(\mathfrak {s})\,d\mathfrak {s}} \phi _{\mathfrak {n}}(\mathbf {x}) \\ & - \sum_{n=\mathcal {N}+1}^{\infty}\frac{ \langle\mathbf{h}(\mathbf {x}),\phi_{\mathfrak {n}}(\mathbf {x}) \rangle}{\int_{0}^{\mathcal {T}_{0}} \mathbf{S}_{\mathfrak {n}}(\mathcal {T}_{0}-\mathfrak {s},\gamma) \mathcal {Q}(\mathfrak {s})\,d\mathfrak {s}} \phi_{\mathfrak {n}}(\mathbf {x}). \end{aligned}$$
This follows from the Parseval identity
$$\begin{aligned} \bigl\Vert \widetilde{f}_{\mathfrak {M},\mathcal {N}}(\cdot)-f(\cdot) \bigr\Vert _{L^{2}(\varOmega)}^{2} ={}& \sum_{\mathfrak {n}=1}^{\mathcal {N}} \biggl[ \frac{ \frac{\pi}{ \mathfrak {M}}\sum_{\mathbf {k}=1}^{\mathfrak {M}}\sigma_{\mathbf {k}}\varepsilon_{\mathbf {k}}\phi_{\mathfrak {n}}(\mathbf {x}_{\mathbf {k}})-\overline{\mathcal{G}}_{\mathfrak {n},\mathfrak {M}} }{\int_{0}^{\mathcal {T}_{0}} \mathbf{S}_{\mathfrak {n}}(\mathcal {T}_{0}-\mathfrak {s},\gamma) \mathcal{Q}(\mathfrak {s})\,d\mathfrak {s}} \biggr]^{2} \\ & +\sum_{\mathfrak {n}=\mathcal {N}+1}^{\infty} \biggl[ \frac{ \langle\mathbf{h}(\mathbf {x}),\phi_{\mathfrak {n}}(\mathbf {x}) \rangle}{\int_{0}^{\mathcal {T}_{0}} \mathbf{S}_{\mathfrak {n}}(\mathcal {T}_{0}-\mathfrak {s},\gamma) \mathcal{Q}(\mathfrak {s})\,d\mathfrak {s}} \biggr]^{2}. \end{aligned}$$
(39)
The fact that
\(\mathbb{E} ( \varepsilon_{j} \varepsilon_{l} )=0\) (
\(l\neq j \)), and
\(\mathbb{E} ( \varepsilon_{j} )=0\) (
\(j = \overline{1,\mathfrak {n}} \)). So we can deduce that
$$\begin{aligned} & \mathbb{E} \bigl\Vert \widetilde{f}_{\mathfrak {M},\mathcal {N}}(\cdot)-f(\cdot) \bigr\Vert _{L^{2}(\varOmega)}^{2} \\ &\quad= \sum_{\mathfrak {n}=\mathcal {N}+1}^{\infty} \biggl[ \frac{ \langle\mathbf{h}(\mathbf {x}),\phi_{\mathfrak {n}}(\mathbf {x}) \rangle}{\int_{0}^{\mathcal {T}_{0}} \mathbf{S}_{\mathfrak {n}}(\mathcal {T}_{0}-\mathfrak {s},\gamma) \mathcal{Q}(\mathfrak {s})\,d\mathfrak {s}} \biggr]^{2} + \sum_{\mathfrak {n}=1}^{\mathcal {N}} \frac{ \frac{\pi^{2}}{\mathfrak {M}^{2}}\sum_{\mathbf {k}=1}^{\mathfrak {M}}\sigma_{\mathbf {k}}^{2} \mathbb{E} \varepsilon_{\mathbf {k}}^{2}+\overline{\mathcal{G}}_{\mathfrak {n},\mathfrak {M}}^{2} }{ [ \int_{0}^{\mathcal {T}_{0}} \mathbf{S}_{\mathfrak {n}}(\mathcal {T}_{0}-\mathfrak {s},\gamma) \mathcal {Q}(\mathfrak {s})\,d\mathfrak {s}]^{2}} \\ &\quad= \mathbb{I}_{1}+\mathbb{I}_{2}. \end{aligned}$$
(40)
We have
$$\begin{aligned} \mathbb{I}_{1}=\sum_{\mathfrak {n}=\mathcal {N}+1}^{\infty} \biggl[ \frac{ \langle\mathbf{h}(\mathbf {x}),\phi_{\mathfrak {n}}(\mathbf {x}) \rangle}{\int _{0}^{\mathcal {T}_{0}} \mathbf{S}_{\mathfrak {n}}(\mathcal {T}_{0}-\mathfrak {s},\gamma) \mathcal{Q}(\mathfrak {s})\,d\mathfrak {s}} \biggr]^{2} . \end{aligned}$$
(41)
By equation (
25), we know that, for
\(\mathfrak {n}\ge1\),
$$\begin{aligned} \bigl\langle f(\mathbf {x}),\phi_{\mathfrak {n}}(\mathbf {x}) \bigr\rangle = \frac{ \langle \mathbf{h}(\mathbf {x}),\phi_{\mathfrak {n}}(\mathbf {x}) \rangle}{\int_{0}^{\mathcal {T}_{0}} \mathbf{S}_{\mathfrak {n}}(\mathcal {T}_{0}-\mathfrak {s},\gamma) \mathcal{Q}(\mathfrak {s})\,d\mathfrak {s}}. \end{aligned}$$
Using the last two equations, we get
$$\begin{aligned} \mathbb{I}_{1}=\sum_{\mathfrak {n}=\mathcal {N}+1}^{\infty} \bigl[ \bigl\langle f(\mathbf {x}),\phi_{\mathfrak {n}}(\mathbf {x}) \bigr\rangle \bigr]^{2}. \end{aligned}$$
By
\(1=\mathfrak {n}^{-2\beta} \mathfrak {n}^{2\beta}\), we can rewrite
\(\mathbb{I}_{1}\) as follows:
$$\begin{aligned} \mathbb{I}_{1}=\sum_{\mathfrak {n}=\mathcal {N}+1}^{\infty} \mathfrak {n}^{-2\beta} \mathfrak {n}^{2\beta} \bigl\vert \bigl\langle f(\mathbf {x}), \phi_{\mathfrak {n}}(\mathbf {x}) \bigr\rangle \bigr\vert ^{2}. \end{aligned}$$
(42)
In the last series (
42), since
\(\mathfrak {n}\geq \mathcal {N}+1 > \mathcal {N}\), we get
\(\mathfrak {n}^{-2\beta}\leq \mathcal {N}^{-2\beta}\). Using the last two observations, we obtain
$$ \mathbb{I}_{1} \le\sum_{\mathfrak {n}=\mathcal {N}+1}^{\infty} \mathcal {N}^{-2\beta} \mathfrak {n}^{2\beta} \bigl\vert \bigl\langle f(\mathbf {x}), \phi_{\mathfrak {n}}(\mathbf {x}) \bigr\rangle \bigr\vert ^{2} = \mathcal {N}^{-2\beta} \sum_{\mathfrak {n}=\mathcal {N}+1}^{\infty} \mathfrak {n}^{2\beta} \bigl\vert \bigl\langle f(\mathbf {x}),\phi_{\mathfrak {n}}(\mathbf {x}) \bigr\rangle \bigr\vert ^{2} = \mathcal {N}^{-2\beta} \widehat{\mathbb{I}}_{1}. $$
(43)
We shall begin with showing that
$$\begin{aligned} \widehat{\mathbb{I}}_{1}=\sum _{\mathfrak {n}=\mathcal {N}+1}^{\infty} \mathfrak {n}^{2\beta} \bigl\vert \bigl\langle f( \mathbf {x}),\phi_{\mathfrak {n}}(\mathbf {x}) \bigr\rangle \bigr\vert ^{2} \leq\sum _{\mathfrak {n}=1}^{\infty} \mathfrak {n}^{2\beta} \bigl\vert \bigl\langle f( \mathbf {x}),\phi_{\mathfrak {n}}(\mathbf {x}) \bigr\rangle \bigr\vert ^{2} = \Vert f \Vert _{\mathcal{H}^{\beta}(\varOmega)}^{2}. \end{aligned}$$
(44)
Using (
43) and (
44), we get
$$\begin{aligned} \mathbb{I}_{1} \leq \mathcal {N}^{-2\beta} \Vert f \Vert _{\mathcal{H}^{\beta}(\varOmega)}^{2} \leq \mathcal {N}^{-2\beta} \mathbf{P}^{2}. \end{aligned}$$
(45)
Recall the definition of
\(\mathbb{I}_{2}\) in equation (
40)
$$\begin{aligned} \mathbb{I}_{2}&=\sum_{\mathfrak {n}=1}^{N} \frac{ \frac{\pi^{2}}{\mathfrak {M}^{2}}\sum_{\mathbf {k}=1}^{\mathfrak {M}}\sigma_{\mathbf {k}}^{2} \mathbb{E} \varepsilon_{\mathbf {k}}^{2}+\overline{\mathcal{G}}_{\mathfrak {n},\mathfrak {M}}^{2} }{ [ \int_{0}^{\mathcal {T}_{0}} \mathbf{S}_{\mathfrak {n}}(\mathcal {T}_{0}-\mathfrak {s},\gamma) \mathcal {Q}(\mathfrak {s})\,d\mathfrak {s}]^{2}} \\ &= \Biggl( \frac{\pi^{2}}{\mathfrak {M}^{2}}\sum_{\mathbf {k}=1}^{\mathfrak {M}} \sigma_{\mathbf {k}}^{2} \mathbb{E} \varepsilon_{\mathbf {k}}^{2}+ \overline{\mathcal{G}}_{\mathfrak {n},\mathfrak {M}}^{2} \Biggr) \sum _{\mathfrak {n}=1}^{\mathcal {N}} \biggl[ \int_{0}^{\mathcal {T}_{0}} \mathbf{S}_{\mathfrak {n}}(\mathcal {T}_{0}-\mathfrak {s}, \gamma) \mathcal{Q}(\mathfrak {s})\,d\mathfrak {s}\biggr]^{-2} . \end{aligned}$$
(46)
We invoke Lemma
3.3 to deduce that
$$\begin{aligned} \bigl\vert \bigl\langle \mathbf{h}(\mathbf {x}),\phi_{\mathfrak {n}}(\mathbf {x}) \bigr\rangle \bigr\vert &= \biggl[ \int_{0}^{\mathcal {T}_{0}} \mathbf{S}_{\mathfrak {n}}(\mathcal {T}_{0}-\mathfrak {s}, \gamma) \mathcal{Q}(\mathfrak {s})\,d\mathfrak {s}\biggr]\bigl\vert \bigl\langle f(\mathbf {x}), \phi_{\mathfrak {n}}( \mathbf {x}) \bigr\rangle \bigr\vert \\ & \le \Vert \mathcal{Q} \Vert _{\infty} \frac{\mathcal {B}_{2}(d,\gamma)}{\mathfrak {n}^{2} } \frac{\mathcal {T}_{0}^{\gamma}}{\gamma} \Vert f \Vert _{L^{2}(\varOmega)}, \end{aligned}$$
(47)
where
\(\sum_{l=1}^{\infty}\frac{1}{l^{2}}=\frac{\pi^{2}}{6}\). We can now combine the results of Lemma
2.2 and equation (
47) to obtain
$$\begin{aligned} \overline{\mathcal{G}}_{\mathfrak {n},\mathfrak {M}} &\le\sum_{l=1}^{\infty} \bigl[ \bigl\langle \mathbf{h}(\mathbf {x}),\phi_{\mathfrak {n}+2l\mathfrak {M}}(\mathbf {x}) \bigr\rangle + \bigl\langle \mathbf{h}(\mathbf {x}),\phi_{-\mathfrak {n}+2l\mathfrak {M}}(\mathbf {x}) \bigr\rangle \bigr] \\ &\le\frac{\mathcal{B}_{2}(d,\gamma) \mathcal {T}_{0}^{\gamma} \Vert \mathcal {Q} \Vert _{\infty} \Vert f \Vert _{L^{2}(\varOmega)} }{\gamma} \Biggl[ \sum_{l=1}^{\infty} \frac{1}{ ( \mathfrak {n}+2l\mathfrak {M})^{2}} + \sum_{l=1}^{\infty} \frac{1}{ ( -\mathfrak {n}+2l\mathfrak {M})^{2}} \Biggr] \\ &\le\frac{\pi^{2}}{6} \frac{\mathcal{B}_{2}(d,\gamma) \mathcal {T}_{0}^{\gamma} \Vert \mathcal{Q} \Vert _{\infty} \Vert f \Vert _{L^{2}(\varOmega)} }{\gamma \mathfrak {M}^{2}} \end{aligned}$$
(48)
$$\begin{aligned} &\le\frac{\pi^{2}}{6} \frac{\mathcal{B}_{2}(d,\gamma) \Vert \mathcal{Q} \Vert _{\infty} \mathbf{P}^{\frac{1}{\beta+1} } \Vert \mathbf{h} \Vert _{L^{2}(\varOmega)}^{\frac{\beta}{\beta +1} } \mathcal {T}_{0}^{\gamma} }{\gamma \mathfrak {M}^{2} \mathcal{Q}_{0}^{ \frac{\beta }{\beta+1}} \mathcal{B}_{1} ^{\frac{\beta}{\beta+1} } (d,\mathcal {T}_{0},\gamma )} . \end{aligned}$$
(49)
Since
\(\sigma_{k} < R_{\mathrm{max}} \), we estimate
\(\mathbb{I}_{2}\) as follows:
$$\begin{aligned} \mathbb{I}_{2} &\le \biggl( \frac{\pi}{ \mathfrak {M}}R_{\mathrm{max}}^{2}+ \frac{\pi^{4}}{36} \frac{\mathcal{B}_{2}^{2}(d,\gamma) \Vert \mathcal {Q} \Vert _{\infty}^{2} \mathbf{P}^{\frac{2}{\beta+1} } \Vert \mathbf{h} \Vert _{L^{2}(\varOmega)}^{\frac{2\beta}{\beta+1} } \mathcal {T}_{0}^{2\gamma} }{\gamma^{2} \mathfrak {M}^{4} \mathcal{Q}_{0}^{ \frac{2\beta}{\beta +1}} \mathcal{B}_{1} ^{\frac{2\beta}{\beta+1} } (d,\mathcal {T}_{0},\gamma)} \biggr) \\& \quad{}\times\sum_{\mathfrak {n}=1}^{\mathcal {N}} \biggl[ \int_{0}^{\mathcal {T}_{0}} \mathbf{S}_{\mathfrak {n}}(\mathcal {T}_{0}-\mathfrak {s}, \gamma) \mathcal{Q}(\mathfrak {s})\,d\mathfrak {s}\biggr]^{-2} \\&\le \biggl( \frac{\pi}{ \mathfrak {M}}R_{\mathrm{max}}^{2}+\frac{\pi^{4}}{36} \frac{\mathcal{B}_{2}^{2}(d,\gamma) \Vert \mathcal{Q} \Vert _{\infty}^{2} \mathbf{P}^{\frac{2}{\beta+1} } \Vert \mathbf{h} \Vert _{L^{2}(\varOmega)}^{\frac{2\beta}{\beta+1} } \mathcal {T}_{0}^{2\gamma} }{\gamma^{2} \mathfrak {M}^{4} \mathcal{Q}_{0}^{ \frac{2\beta}{\beta+1}} \mathcal {B}_{1} ^{\frac{2\beta}{\beta+1} } (d,\mathcal {T}_{0},\gamma)} \biggr) \sum_{\mathfrak {n}=1}^{\mathcal {N}} \frac{\mathfrak {n}^{4} }{\mathcal{Q}_{0}^{2} \mathcal{B}_{1} ^{2} (d,\mathcal {T}_{0},\gamma) } \\&\le \biggl( \frac{\pi}{ \mathfrak {M}\mathcal{Q}_{0}^{2} \mathcal{B}_{1} ^{2} (d,\mathcal {T}_{0},\gamma) }R_{\mathrm{max}}^{2}+\frac{\pi^{4}}{36} \frac{\mathcal{B}_{2}^{2}(d,\gamma) \Vert \mathcal{Q} \Vert _{\infty}^{2} \mathbf{P}^{\frac{2}{\beta+1} } \Vert \mathbf{h} \Vert _{L^{2}(\varOmega)}^{\frac{2\beta}{\beta+1} } \mathcal {T}_{0}^{2\gamma} }{\gamma^{2} \mathfrak {M}^{4} \mathcal{Q}_{0}^{ \frac{4\beta+2}{\beta+1}} \mathcal{B}_{1} ^{\frac{4\beta+2}{\beta+1} } (d,\mathcal {T}_{0},\gamma)} \biggr) \mathcal {N}^{5}. \end{aligned}$$
(50)
Combining equation (
45) with equation (
50), we obtain
$$\begin{aligned} &\mathbb{E} \Vert \widetilde{f}_{\mathfrak {M},N}-f \Vert _{L^{2}(\varOmega)}^{2} \\&\quad\leq \biggl( \frac{\pi}{ \mathfrak {M}\mathcal{Q}_{0}^{2} \mathcal{B}_{1} ^{2} (d,\mathcal {T}_{0},\gamma) }R_{\mathrm{max}}^{2}+\frac{\pi^{4}}{36} \frac{\mathcal{B}_{2}^{2}(d,\gamma) \Vert \mathcal{Q} \Vert _{\infty}^{2} \mathbf{P}^{\frac{2}{\beta+1} } \Vert \mathbf{h} \Vert _{L^{2}(\varOmega)}^{\frac{2\beta}{\beta +1} } \mathcal {T}_{0}^{2\gamma} }{\gamma^{2} \mathfrak {M}^{4} \mathcal{Q}_{0}^{ \frac{4\beta +2}{\beta+1}} \mathcal{B}_{1} ^{\frac{4\beta+2}{\beta+1} } (d,\mathcal {T}_{0},\gamma)} \biggr) \mathcal {N}^{5} + \mathcal {N}^{-2\beta} \mathbf{P}^{2}. \end{aligned}$$
(51)
It is shown that our main results are stated and proved. □