4. Error analysis

In last section we have proved the convergence property of the regularized Fourier series under the condition <sup>f</sup> <sup>T</sup> <sup>∈</sup>L<sup>1</sup> <sup>R</sup><sup>2</sup> � � . In this section, we give the error analysis of the regularized Fourier series according to the L<sup>2</sup> -norm for the functions <sup>f</sup> <sup>T</sup> <sup>∈</sup> <sup>L</sup><sup>2</sup> <sup>R</sup><sup>2</sup> � �. The bound of the variance of the regularized Fourier series is presented.

By Lemma 3.5, we have next lemma.

### Lemma 4.1

$$\sum\_{n\_1=-\infty}^{\infty} \sum\_{n\_2=-\infty}^{\infty} \left| \frac{\eta(n\_1 H\_1, n\_2 H\_2)}{[1 + 2\pi a + 2\pi a (n\_1 H\_1)^2][1 + 2\pi a + 2\pi a (n\_2 H\_2)^2]} \right|^2 = O(\delta^2) + O\left(\frac{\delta^2}{a}\right)$$

for δ ! þ0 and α ! þ0, where η and δ are given in Eq. (4) and Eq. (5) in Section 1.

Theorem 4.1 Suppose <sup>f</sup> <sup>T</sup> <sup>∈</sup>L<sup>2</sup> <sup>R</sup><sup>2</sup> � � is band-limited. If we choose <sup>α</sup> <sup>¼</sup> α δð Þ such that α δð Þ! 0 and <sup>δ</sup><sup>2</sup> <sup>=</sup>α δð Þ! 0 as <sup>δ</sup> ! 0, then ^<sup>f</sup> <sup>α</sup>ð Þ! <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> ^<sup>f</sup> <sup>T</sup>ð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> in L2 ½ �� � �Ω1; Ω<sup>1</sup> ½ � Ω2; Ω<sup>2</sup> as δ ! 0.

Proof.

$$\begin{split} &\hat{f}\_{a}(\boldsymbol{w}\_{1},\boldsymbol{w}\_{2}) - \hat{f}\_{T}(\boldsymbol{w}\_{1},\boldsymbol{w}\_{2}) \\ &= H\_{1}H\_{2} \sum\_{n\_{1}=-\infty}^{\infty} \sum\_{n\_{2}=-\infty}^{\infty} \frac{\left[\hat{f}\_{T}(n\_{1}H\_{1},n\_{2}H\_{2}) + \eta(n\_{1}H\_{1},n\_{2}H\_{2})\right]e^{\hat{\mu}\_{1}H\_{1}\boldsymbol{w}\_{1} + \hat{\mu}\_{2}H\_{2}\boldsymbol{w}\_{2}}}{\left[1+2\pi a + 2\pi a(n\_{1}H\_{1})^{2}\right]\left[1+2\pi a + 2\pi a(n\_{2}H\_{2})^{2}\right]}P\_{2}(\boldsymbol{w}\_{1},\boldsymbol{w}\_{2}) - \hat{f}\_{T}(\boldsymbol{w}\_{1},\boldsymbol{w}\_{2}) \\ &= -H\_{1}H\_{2} \sum\_{n\_{1}=-\infty}^{\infty} \sum\_{n\_{2}=-\infty}^{\infty} \frac{4\pi a + 2\pi a(n\_{1}H\_{1})^{2} + 2\pi a(n\_{2}H\_{2})^{2} + \left(2\pi a + 2\pi a(n\_{1}H\_{1})^{2}\right)\left(2\pi a + 2\pi a(n\_{2}H\_{2})^{2}\right)}{\left[1 + 2\pi a + 2\pi a(n\_{1}H\_{1})^{2}\right]\left[1 + 2\pi a + 2\pi a(n\_{2}H\_{2})^{2}\right]} \end{split}$$

$$\begin{split} \left. \begin{gathered} \mathcal{f}\_{T}(n\_{1}H\_{1},n\_{2}H\_{2})e^{in\_{1}H\_{1}\omega\_{1} + in\_{2}H\_{2}\omega\_{2}}P\_{\mathcal{Q}}(\mathbf{o}\_{1},\mathbf{o}\_{2})\\ + \, \_{1}^{2}H\_{2} \sum\_{n\_{1} = -\infty}^{\infty} \sum\_{n\_{2} = -\infty}^{\infty} \frac{\eta(n\_{1}H\_{1},n\_{2}H\_{2})}{\left[1 + 2\pi\alpha + 2\pi\alpha(n\_{1}H\_{1})^{2}\right] \left[1 + 2\pi\alpha + 2\pi\alpha(n\_{2}H\_{2})^{2}\right]} \end{gathered} \right| \\ \begin{split} \, e^{in\_{1}H\_{1}\alpha\_{1} + in\_{2}H\_{2}\alpha\_{2}}P\_{\mathcal{Q}}(\mathbf{o}\_{1},\mathbf{o}\_{2}). \end{split} \end{split}$$

∑ ∞ n1¼�∞

�

�

�

where

∑ ∣n1∣>N

�

and

≤ ∑ ∣n1∣≥N

as α ! 0.

55

∑ ∞ n2¼�∞ 4πα þ 2παð Þ n1H<sup>1</sup>

� f <sup>T</sup>ð Þ n1H1; n2H<sup>2</sup>

4πα þ 2παð Þ n1H<sup>1</sup>

� f <sup>T</sup>ð Þ n1H1; n2H<sup>2</sup>

4πα þ 2παð Þ n1H<sup>1</sup>

� f <sup>T</sup>ð Þ n1H1; n2H<sup>2</sup>

∑or∣n2∣>N 4πα þ 2παð Þ n1H<sup>1</sup>

� f <sup>T</sup>ð Þ n1H1; n2H<sup>2</sup>

4πα þ 2παð Þ n1H<sup>1</sup>

� � �

� �

∑or∣n2∣≥N

� �

� �

� �

� <sup>2</sup> <sup>¼</sup> <sup>∑</sup> ∣n1∣ ≤ N

DOI: http://dx.doi.org/10.5772/intechopen.81542

� 2 þ ∑ ∣n1∣>N

� 2 ,

� 2

Therefore, ^<sup>f</sup> <sup>α</sup>ð Þ� <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> ^<sup>f</sup> <sup>T</sup>ð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup>

bias ^<sup>f</sup> <sup>T</sup>ð Þ� <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> <sup>E</sup> ^<sup>f</sup> <sup>α</sup>ð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup>

�

<sup>2</sup> <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

f <sup>T</sup>ð Þ n1H1; n2H<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

1 þ 2πα þ 2παð Þ n1H<sup>1</sup>

�

h i ! 0 in <sup>L</sup><sup>2</sup>

Var ^<sup>f</sup> <sup>α</sup>ð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup>

� �

1 þ 2πα þ 2παð Þ n1H<sup>1</sup>

� 2 < ε

∑ ∣n1∣ ≤ N

<sup>2</sup> h i <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

f <sup>T</sup>ð Þ n1H1; n2H<sup>2</sup>

� � � 2 <sup>L</sup><sup>2</sup> ! 0.

Theorem 4.2 Suppose <sup>f</sup> <sup>T</sup> <sup>∈</sup>L<sup>2</sup> <sup>R</sup><sup>2</sup> � � is band-limited. If the noise in Eq. (4) is white noise such that <sup>E</sup>½ηð Þ <sup>n</sup>1H1; <sup>n</sup>2H<sup>2</sup> � ¼ 0 and Var½ηð Þ <sup>n</sup>1H1; <sup>n</sup>2H<sup>2</sup> � ¼ <sup>σ</sup>2, then the

h i <sup>¼</sup> <sup>O</sup> <sup>σ</sup><sup>2</sup> � � <sup>þ</sup> <sup>O</sup> <sup>σ</sup><sup>2</sup>

� �

1 þ 2πα þ 2παð Þ n1H<sup>1</sup>

1 þ 2πα þ 2παð Þ n1H<sup>1</sup>

∑ or∣n2∣>N

1 þ 2πα þ 2παð Þ n1H<sup>1</sup>

∑ and ∣n2∣ ≤ N

Computation of Two-Dimensional Fourier Transforms for Noisy Band-Limited Signals

<sup>2</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>1H<sup>1</sup>

<sup>2</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>1H<sup>1</sup>

<sup>2</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>1H<sup>1</sup>

<sup>2</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>1H<sup>1</sup>

∑ and ∣n2∣ ≤ N

<sup>2</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>1H<sup>1</sup>

� <sup>2</sup> ! <sup>0</sup>

<sup>2</sup> h i <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

<sup>2</sup> h i <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

<sup>2</sup> h i <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

<sup>2</sup> h i <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

<sup>2</sup> � � <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

<sup>2</sup> � � <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

<sup>2</sup> � � <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

<sup>2</sup> � � <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

<sup>2</sup> � � <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

½ �� � �Ω1; Ω<sup>1</sup> ½ � Ω2; Ω<sup>2</sup> as α ! 0 and

=α � �

<sup>2</sup> h i

<sup>2</sup> h i

<sup>2</sup> h i

<sup>2</sup> h i

<sup>2</sup> h i

<sup>2</sup> � �

<sup>2</sup> � �

<sup>2</sup> � �

<sup>2</sup> � �

<sup>2</sup> � �

Let

$$\begin{split} \mathbf{S}(\boldsymbol{\omega}\_{1},\boldsymbol{\alpha}\_{2}) & \coloneqq \sum\_{n\_{1}=-\infty}^{\infty} \sum\_{n\_{2}=-\infty}^{\infty} \\ \frac{4\pi\alpha + 2\pi\alpha (n\_{1}H\_{1})^{2} + 2\pi\alpha (n\_{2}H\_{2})^{2} + \left(2\pi\alpha + 2\pi\alpha (n\_{1}H\_{1})^{2}\right) \left(2\pi\alpha + 2\pi\alpha (n\_{2}H\_{2})^{2}\right)}{\left[1 + 2\pi\alpha + 2\pi\alpha (n\_{1}H\_{1})^{2}\right] \left[1 + 2\pi\alpha + 2\pi\alpha (n\_{2}H\_{2})^{2}\right]} \\ & \qquad \cdot \int\_{T} (n\_{1}H\_{1}, n\_{2}H\_{2}) e^{in\_{1}H\_{1}\alpha\_{1} + in\_{2}H\_{2}\alpha\_{2}} P\_{\mathcal{Q}}(\boldsymbol{\alpha}\_{1},\boldsymbol{\alpha}\_{2}). \end{split}$$

Then

$$\left\|\hat{f}\_{\boldsymbol{a}}(\boldsymbol{\omega}\_{1},\boldsymbol{\omega}\_{2})-\hat{f}\_{T}(\boldsymbol{\omega}\_{1},\boldsymbol{\omega}\_{2})\right\|\_{L^{2}}^{2}\leq 2H\_{1}^{2}H\_{2}^{2}\left\|\mathbf{S}(\boldsymbol{\omega}\_{1},\boldsymbol{\omega}\_{2})\right\|^{2}+2H\_{1}^{2}H\_{2}^{2}$$

$$\left\|\sum\_{\boldsymbol{n}\_{1}=-\infty}^{\infty}\sum\_{n\_{2}=-\infty}^{\infty}\frac{\eta(n\_{1}H\_{1},n\_{2}H\_{2})}{\left[1+2\pi\alpha+2\pi\alpha(n\_{1}H\_{1})^{2}\right]\left[1+2\pi\alpha+2\pi\alpha(n\_{2}H\_{2})^{2}\right]}e^{\boldsymbol{m}\_{1}H\_{1}\boldsymbol{\omega}\_{1}+\dot{m}\_{2}H\_{2}\boldsymbol{m}\_{2}}P\_{\mathcal{Q}}(\boldsymbol{\omega}\_{1},\boldsymbol{\omega}\_{2})\right\|^{2},$$

where

$$\begin{aligned} & \left\| \sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} \frac{\eta(n\_1 H\_1, n\_2 H\_2)}{\left[1 + 2\pi\alpha + 2\pi\alpha (n\_1 H\_1)^2\right] \left[1 + 2\pi\alpha + 2\pi\alpha (n\_2 H\_2)^2\right]} e^{in\_1 H\_1 \alpha\_1 + n\_2 H\_2 \alpha\_2} P\_{\mathcal{Q} \left(\mathfrak{w}\_1, \mathfrak{w}\_2\right)} \right\|^2 \\ &= \sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} \left| \frac{\eta(n\_1 H\_1, n\_2 H\_2)}{\left[1 + 2\pi\alpha + 2\pi\alpha (n\_1 H\_1)^2\right] \left[1 + 2\pi\alpha + 2\pi\alpha (n\_2 H\_2)^2\right]} \right|^2 = O\left(\frac{\delta^2}{a}\right) \end{aligned}$$

by Lemma 4.1 and

$$\left\|\mathbf{S}(\alpha\_{1},\alpha\_{2})\right\|^{2} = \sum\_{n\_{1}=-\infty}^{\infty} \sum\_{n\_{2}=-\infty}^{\infty}$$

$$\frac{4\pi a + 2\pi a(n\_{1}H\_{1})^{2} + 2\pi a(n\_{2}H\_{2})^{2} + \left(2\pi a + 2\pi a(n\_{1}H\_{1})^{2}\right)\left(2\pi a + 2\pi a(n\_{2}H\_{2})^{2}\right)}{\left[1 + 2\pi a + 2\pi a(n\_{1}H\_{1})^{2}\right]\left[1 + 2\pi a + 2\pi a(n\_{2}H\_{2})^{2}\right]}$$

$$\left|f\_{T}(n\_{1}H\_{1},n\_{2}H\_{2})\right|^{2}.$$

For every ε>0, there exists N>0 such that

$$\sum\_{|n\_1| \ge N} \sum\_{\substack{\sigma \colon |n\_2| \ge N}} \left| f\_{\,^T}(n\_1 H\_1, n\_2 H\_2) \right|^2 < \epsilon,$$

since

Computation of Two-Dimensional Fourier Transforms for Noisy Band-Limited Signals DOI: http://dx.doi.org/10.5772/intechopen.81542

∑ ∞ n1¼�∞ ∑ ∞ n2¼�∞ 4πα þ 2παð Þ n1H<sup>1</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>1H<sup>1</sup> <sup>2</sup> � � <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup> <sup>2</sup> � � 1 þ 2πα þ 2παð Þ n1H<sup>1</sup> <sup>2</sup> h i <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup> <sup>2</sup> h i � f <sup>T</sup>ð Þ n1H1; n2H<sup>2</sup> � � � � <sup>2</sup> <sup>¼</sup> <sup>∑</sup> ∣n1∣ ≤ N ∑ and ∣n2∣ ≤ N 4πα þ 2παð Þ n1H<sup>1</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>1H<sup>1</sup> <sup>2</sup> � � <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup> <sup>2</sup> � � 1 þ 2πα þ 2παð Þ n1H<sup>1</sup> <sup>2</sup> h i <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup> <sup>2</sup> h i � f <sup>T</sup>ð Þ n1H1; n2H<sup>2</sup> � � � � 2 þ ∑ ∣n1∣>N ∑ or∣n2∣>N 4πα þ 2παð Þ n1H<sup>1</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>1H<sup>1</sup> <sup>2</sup> � � <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup> <sup>2</sup> � � 1 þ 2πα þ 2παð Þ n1H<sup>1</sup> <sup>2</sup> h i <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup> <sup>2</sup> h i � f <sup>T</sup>ð Þ n1H1; n2H<sup>2</sup> � � � � 2 ,

where

�<sup>f</sup> <sup>T</sup>ð Þ <sup>n</sup>1H1; <sup>n</sup>2H<sup>2</sup> <sup>e</sup>in1H1ω1þin2H2ω2PΩð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup>

<sup>e</sup>in1H1ω1þin2H2ω2PΩð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> :

n1¼�∞

�<sup>f</sup> <sup>T</sup>ð Þ <sup>n</sup>1H1; <sup>n</sup>2H<sup>2</sup> <sup>e</sup>in1H1ω1þin2H2ω2PΩð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> :

<sup>2</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>1H<sup>1</sup> <sup>2</sup> � �

1H<sup>2</sup>

1 þ 2πα þ 2παð Þ n2H<sup>2</sup> <sup>2</sup> h i <sup>e</sup>

1 þ 2πα þ 2παð Þ n2H<sup>2</sup> <sup>2</sup> h i <sup>e</sup>

1 þ 2πα þ 2παð Þ n2H<sup>2</sup> <sup>2</sup> h i

> ∑ ∞ n2¼�∞

1 þ 2πα þ 2παð Þ n2H<sup>2</sup> <sup>2</sup> h i

> � 2 < ϵ,

� 2 :

<sup>2</sup> <sup>¼</sup> <sup>∑</sup> ∞ n1¼�∞

� f <sup>T</sup>ð Þ n1H1; n2H<sup>2</sup>

� �

<sup>2</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>1H<sup>1</sup> <sup>2</sup> � �

f <sup>T</sup>ð Þ n1H1; n2H<sup>2</sup>

� �

1 þ 2πα þ 2παð Þ n1H<sup>1</sup> <sup>2</sup> h i

<sup>S</sup>ð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> ≔ ∑∞

� � � 2 <sup>L</sup><sup>2</sup> <sup>≤</sup>2H<sup>2</sup>

ηð Þ n1H1; n2H<sup>2</sup>

ηð Þ n1H1; n2H<sup>2</sup>

ηð Þ n1H1; n2H<sup>2</sup>

k k Sð Þ ω1; ω<sup>2</sup>

�

∑or∣n2∣≥N �

ηð Þ n1H1; n2H<sup>2</sup>

∑ ∞ n2¼�∞

1 þ 2πα þ 2παð Þ n2H<sup>2</sup> <sup>2</sup> h i

<sup>2</sup>k k Sð Þ ω1; ω<sup>2</sup>

1 þ 2πα þ 2παð Þ n2H<sup>2</sup> <sup>2</sup> h i

> <sup>2</sup> <sup>þ</sup> <sup>2</sup>H<sup>2</sup> 1H<sup>2</sup> 2

� � � � � �

2

<sup>¼</sup> <sup>O</sup> <sup>δ</sup><sup>2</sup> α � �

2πα þ 2παð Þ n2H<sup>2</sup> <sup>2</sup> � �

2πα þ 2παð Þ n2H<sup>2</sup> <sup>2</sup> � �

in1H1ω1þin2H2ω2PΩð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup>

in1H1ω1þin2H2ω2PΩð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup>

� � � � � �

2 ,

� � � � � �

2

þH1H<sup>2</sup> ∑

4πα þ 2παð Þ n1H<sup>1</sup>

� � �

Let

Then

∑ ∞ n1¼�∞

∑ ∞ n1¼�∞

¼ ∑ ∞ n1¼�∞

where

� � � � � �

� � � � � �

∑ ∞ n2¼�∞

∑ ∞ n2¼�∞

> ∑ ∞ n2¼�∞

by Lemma 4.1 and

4πα þ 2παð Þ n1H<sup>1</sup>

since

54

� � � � � �

∞ n1¼�∞

Recent Advances in Integral Equations

∑ ∞ n2¼�∞

<sup>2</sup> <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

^<sup>f</sup> <sup>α</sup>ð Þ� <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> ^<sup>f</sup> <sup>T</sup>ð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>1H<sup>1</sup> <sup>2</sup> h i

1 þ 2πα þ 2παð Þ n1H<sup>1</sup> <sup>2</sup> h i

1 þ 2πα þ 2παð Þ n1H<sup>1</sup> <sup>2</sup> h i

<sup>2</sup> <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

∑ ∣n1∣≥N

For every ε>0, there exists N>0 such that

1 þ 2πα þ 2παð Þ n1H<sup>1</sup> <sup>2</sup> h i

1 þ 2πα þ 2παð Þ n1H<sup>1</sup> <sup>2</sup> h i

$$\begin{split} &\sum\_{|n\_1|>N} \sum\_{\sigma \mid n\_2 \mid \,\sigma} \\ &4\pi\alpha + 2\pi\alpha (n\_1H\_1)^2 + 2\pi\alpha (n\_2H\_2)^2 + \left(2\pi\alpha + 2\pi\alpha (n\_1H\_1)^2\right) \left(2\pi\alpha + 2\pi\alpha (n\_2H\_2)^2\right) \\ &\quad \left[1 + 2\pi\alpha + 2\pi\alpha (n\_1H\_1)^2\right] \left[1 + 2\pi\alpha + 2\pi\alpha (n\_2H\_2)^2\right] \\ &\quad \cdot \left|f\_T(n\_1H\_1, n\_2H\_2)\right|^2 \\ &\leq \sum\_{|n\_1|\geq N} \sum\_{\sigma \mid n\_2\geq N} \left|f\_T(n\_1H\_1, n\_2H\_2)\right|^2 < \varepsilon \end{split}$$

and

$$\sum\_{\left|n\_1\right| \le N} \sum\_{\left|m\_2\right| \le N}$$

$$\frac{4\pi a + 2\pi a (n\_1 H\_1)^2 + 2\pi a (n\_2 H\_2)^2 + \left(2\pi a + 2\pi a (n\_1 H\_1)^2\right) \left(2\pi a + 2\pi a (n\_2 H\_2)^2\right)}{\left[1 + 2\pi a + 2\pi a (n\_1 H\_1)^2\right] \left[1 + 2\pi a + 2\pi a (n\_2 H\_2)^2\right]}$$

$$\left|f\_T(n\_1 H\_1, n\_2 H\_2)\right|^2 \to 0$$

� � � 2 <sup>L</sup><sup>2</sup> ! 0.

as α ! 0. Therefore, ^<sup>f</sup> <sup>α</sup>ð Þ� <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> ^<sup>f</sup> <sup>T</sup>ð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> � � �

Theorem 4.2 Suppose <sup>f</sup> <sup>T</sup> <sup>∈</sup>L<sup>2</sup> <sup>R</sup><sup>2</sup> � � is band-limited. If the noise in Eq. (4) is white noise such that <sup>E</sup>½ηð Þ <sup>n</sup>1H1; <sup>n</sup>2H<sup>2</sup> � ¼ 0 and Var½ηð Þ <sup>n</sup>1H1; <sup>n</sup>2H<sup>2</sup> � ¼ <sup>σ</sup>2, then the bias ^<sup>f</sup> <sup>T</sup>ð Þ� <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> <sup>E</sup> ^<sup>f</sup> <sup>α</sup>ð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> h i ! 0 in <sup>L</sup><sup>2</sup> ½ �� � �Ω1; Ω<sup>1</sup> ½ � Ω2; Ω<sup>2</sup> as α ! 0 and

$$\text{Var}\left[\hat{f}\_a(\mathbf{o}\_1, \mathbf{o}\_2)\right] = O\left(\sigma^2\right) + O\left(\sigma^2/a\right)$$

$$\text{if } a(\sigma) \to 0 \text{ and } \sigma^2/a(\sigma) \to 0 \text{ as } \sigma \to 0.$$

Proof. We can calculate

$$\begin{aligned} & \left\| \hat{f}\_{T}(\mathbf{u}\_{1},\mathbf{u}\_{2}) - E \left[ \hat{f}\_{a}(\mathbf{u}\_{1},\mathbf{u}\_{2}) \right] \right\|\_{L^{2}}^{2} = H\_{1}^{2}H\_{2}^{2} \quad \cdot \sum\_{n\_{1}=-\infty}^{\infty} \sum\_{n\_{2}=-\infty}^{\infty} \\ & \left\| \frac{4\pi a + 2\pi a(n\_{1}H\_{1})^{2} + 2\pi a(n\_{2}H\_{2})^{2} + \left(2\pi a + 2\pi a(n\_{1}H\_{1})^{2}\right) \left(2\pi a + 2\pi a(n\_{2}H\_{2})^{2}\right)}{\left[1 + 2\pi a + 2\pi a(n\_{1}H\_{1})^{2}\right] \left[1 + 2\pi a + 2\pi a(n\_{2}H\_{2})^{2}\right]} \right\|\_{L^{2}} \\ & \left\| \cdot f\_{T}(n\_{1}H\_{1},n\_{2}H\_{2}) \right\|^{2} \end{aligned}$$

and

$$\operatorname{Var}\left[\hat{f}\_a(\mathbf{u}\_1, \mathbf{u}\_2)\right] = \sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} \frac{\sigma^2}{\left[1 + 2\pi a + 2\pi a (n\_1 H\_1)^2\right]^2 \left[1 + 2\pi a + 2\pi a (n\_2 H\_2)^2\right]^2}.$$

By the proof of Theorem 4.1, we can see that ^<sup>f</sup> <sup>T</sup>ð Þ� <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> <sup>E</sup> ^<sup>f</sup> <sup>α</sup>ð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> h i ! 0 in L2 ½ �� � �Ω1; <sup>Ω</sup><sup>1</sup> ½ � <sup>Ω</sup>2; <sup>Ω</sup><sup>2</sup> as <sup>α</sup> ! 0 and Var ^<sup>f</sup> <sup>α</sup>ð Þ ½ � �ω1; <sup>ω</sup><sup>1</sup> h i <sup>¼</sup> <sup>O</sup> <sup>σ</sup><sup>2</sup> ð Þþ <sup>O</sup> <sup>σ</sup><sup>2</sup> ð Þ <sup>=</sup><sup>α</sup> .
