3. The regularized Fourier series

Based on the one-dimensional regularized Fourier series in [12], we construct the two-dimensional regularized Fourier series:

$$\begin{aligned} \hat{f}\_a(\mathbf{u}\_1, \mathbf{u}\_2) &= \\ H\_1 H\_2 \sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} \frac{f(n\_1 H\_1, n\_2 H\_2) e^{in\_1 H\_1 \mathbf{u}\_1 + in\_2 H\_2 \mathbf{u}\_2}}{\left[1 + 2\pi a + 2\pi a (n\_1 H\_1)^2\right] \left[1 + 2\pi a + 2\pi a (n\_2 H\_2)^2\right]} P\_{\mathcal{Q}}(\mathbf{u}\_1, \mathbf{u}\_2), \end{aligned} \tag{8}$$

$$F\left[\frac{1}{1+2\pi a + 2\pi a t^2} \frac{\sin \Omega(t - nH)}{\Omega(t - nH)}\right] = \frac{H}{1+2\pi a + 2\pi a (nH)^2} e^{inHa} - \frac{H}{4\pi a a} (-1)^a \left[\frac{e^{a(a-\Omega)}}{a-inH} + \frac{e^{-a(a+\Omega)}}{a+inH}\right],\tag{9}$$

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

Proof. By the sampling theorem

where f nð Þ <sup>1</sup>H1; n2H<sup>2</sup> is given in (4). We will give the convergence property of

<sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ nH <sup>2</sup> <sup>e</sup>

<sup>2</sup><sup>π</sup> <sup>F</sup> <sup>1</sup>

∞ð

�∞ e �a uj je

<sup>4</sup>πa<sup>α</sup> <sup>e</sup>

� �

� <sup>H</sup> <sup>1</sup> <sup>4</sup>πa<sup>α</sup> <sup>e</sup>

inH<sup>ω</sup>ð Þ �<sup>1</sup> <sup>n</sup> eað Þ� <sup>ω</sup>�<sup>Ω</sup> inH<sup>ω</sup>

<sup>4</sup>πa<sup>α</sup> ð Þ �<sup>1</sup> <sup>n</sup> <sup>e</sup><sup>a</sup>ð Þ <sup>ω</sup>�<sup>Ω</sup>

2 2

ein1H1ω<sup>1</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>

ein2H2ω<sup>2</sup> <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>

> eað Þ <sup>ω</sup>1�Ω<sup>1</sup> a � in1H<sup>1</sup>

inHω

1

1 þ 2πα þ 2παt<sup>2</sup> � �

> ð0 ω�Ω e

<sup>a</sup> � inH <sup>þ</sup>

� �

ein1H1ω1þin2H2ω<sup>2</sup> ½1 þ 2πα þ 2πα n1H1Þ<sup>2</sup> ð �½1 þ 2πα þ 2πα n2H2Þ<sup>2</sup> ð �

> ð Þ �<sup>1</sup> <sup>n</sup><sup>2</sup> 4πaα

> ð Þ �<sup>1</sup> <sup>n</sup><sup>1</sup> 4πaα

<sup>þ</sup> <sup>e</sup>�að Þ <sup>ω</sup>1þΩ<sup>1</sup> a þ in1H<sup>1</sup>

� �

" # � �

" # � �

<sup>a</sup> � inH <sup>þ</sup>

<sup>a</sup> <sup>þ</sup> inH � <sup>e</sup>�ð Þ <sup>a</sup>þinH ð Þ <sup>ω</sup>þ<sup>Ω</sup>

inH<sup>ω</sup> � <sup>H</sup>

<sup>4</sup>πa<sup>α</sup> ð Þ �<sup>1</sup> <sup>n</sup> eað Þ <sup>ω</sup>�<sup>Ω</sup>

<sup>∗</sup><sup>F</sup> sin <sup>Ω</sup>ð Þ <sup>t</sup> � nH Ωð Þ t � nH � �

> ð<sup>ω</sup>þ<sup>Ω</sup> 0 e

� �

� �

inHð Þ <sup>ω</sup>�<sup>u</sup> <sup>1</sup>½ � <sup>ω</sup>�Ω;ωþ<sup>Ω</sup> ð Þ <sup>u</sup> du

au�inHudu <sup>þ</sup>

<sup>a</sup> � inH <sup>þ</sup>

e�að Þ� <sup>ω</sup>þ<sup>Ω</sup> inH<sup>ω</sup> a þ inH

:

eað Þ <sup>ω</sup>2�Ω<sup>2</sup> a � in2H<sup>2</sup>

e<sup>a</sup>ð Þ <sup>ω</sup>1�Ω<sup>1</sup> a � in1H<sup>1</sup>

> � eað Þ <sup>ω</sup>2�Ω<sup>2</sup> a � in2H<sup>2</sup>

<sup>þ</sup> <sup>e</sup>�að Þ <sup>ω</sup>2þΩ<sup>2</sup> a þ in2H<sup>2</sup>

<sup>þ</sup> <sup>e</sup>�að Þ <sup>ω</sup>1þΩ<sup>1</sup> a þ in1H<sup>1</sup>

� �

<sup>þ</sup> <sup>e</sup>�að Þ <sup>ω</sup>2þΩ<sup>2</sup> a þ in2H<sup>2</sup>

(10)

:

a þ inH

inH<sup>ω</sup> eð Þ <sup>a</sup>�inH ð Þ <sup>ω</sup>�<sup>Ω</sup>

� �

e�að Þ <sup>ω</sup>þ<sup>Ω</sup> a þ inH <sup>a</sup> � inH <sup>þ</sup> <sup>e</sup>�að Þ <sup>ω</sup>þ<sup>Ω</sup>

� �

a þ inH

�au�inHudu

e�ð Þ <sup>a</sup>þinH ð Þ <sup>ω</sup>þ<sup>Ω</sup> a þ inH

(9)

,

<sup>¼</sup> <sup>H</sup>

¼ 1

4πaα

�a∣u∣�inHudu <sup>¼</sup> <sup>H</sup> <sup>1</sup>

1 a þ inH

<sup>a</sup> � inH <sup>þ</sup>

<sup>a</sup> � inH � <sup>e</sup>ð Þ <sup>a</sup>�inH ð Þ <sup>ω</sup>�<sup>Ω</sup>

� �

<sup>4</sup>πa<sup>α</sup> <sup>e</sup>

Lemma 3.2 For any band-limited function g tð Þ <sup>1</sup>; t<sup>2</sup> and

� �

ð Þ �<sup>1</sup> <sup>n</sup>1þn<sup>2</sup> ð Þ <sup>4</sup>πa<sup>α</sup> <sup>2</sup>

the regularized Fourier series in this section.

sin Ωð Þ t � nH Ωð Þ t � nH

ffiffiffiffiffiffiffiffiffiffi 1þ2πα 2πα q

� �

ð<sup>ω</sup>þ<sup>Ω</sup> ω�Ω e

inH<sup>ω</sup> 1

inH<sup>ω</sup> 1

einH<sup>ω</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> ð Þ nH <sup>2</sup> � <sup>H</sup> <sup>1</sup>

<sup>a</sup> � inH <sup>þ</sup>

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

ð Þ ω1; ω<sup>2</sup> ∈½ �� � �Ω1; Ω<sup>1</sup> ½ � Ω2; Ω<sup>2</sup>

1 þ 2πα þ 2παt

∑ ∞ n2¼�∞

∑ ∞ n2¼�∞

∑ ∞ n2¼�∞

∑ ∞ n2¼�∞

g tð Þ <sup>1</sup>; <sup>t</sup><sup>2</sup> eit1ω1þit2ω2dt1dt<sup>2</sup>

g nð Þ <sup>1</sup>H1; n2H<sup>2</sup>

g nð Þ <sup>1</sup>H1; n2H<sup>2</sup>

g nð Þ <sup>1</sup>H1; n2H<sup>2</sup>

g nð Þ <sup>1</sup>H1; n2H<sup>2</sup>

2 1 � � <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα<sup>t</sup>

.

sin Ωð Þ t � nH Ωð Þ t � nH

�aj j <sup>ω</sup> <sup>∗</sup> He<sup>i</sup>ωnHPΩð Þ <sup>ω</sup> � � <sup>¼</sup> <sup>H</sup> <sup>1</sup>

Lemma 3.1

1 þ 2πα þ 2παt<sup>2</sup>

where a ≔

1 þ 2πα þ 2παt<sup>2</sup>

inHω

Proof.

<sup>F</sup> <sup>1</sup>

¼ 1 2π 1 <sup>2</sup>a<sup>α</sup> <sup>e</sup>

<sup>¼</sup> <sup>H</sup> <sup>1</sup> <sup>4</sup>πa<sup>α</sup> <sup>e</sup>

<sup>¼</sup> <sup>H</sup> <sup>1</sup> <sup>4</sup>πa<sup>α</sup> <sup>e</sup>

<sup>¼</sup> <sup>H</sup> <sup>1</sup> <sup>4</sup>πa<sup>α</sup> <sup>e</sup>

<sup>¼</sup> <sup>H</sup> <sup>1</sup> 2πα

ð<sup>∞</sup> �∞ ð<sup>∞</sup> �∞

¼ H1H<sup>2</sup> ∑

�H1H<sup>2</sup> ∑ ∞ n1¼�∞

�H1H<sup>2</sup> ∑ ∞ n1¼�∞

þH1H<sup>2</sup> ∑ ∞ n1¼�∞

48

∞ n1¼�∞

<sup>¼</sup> <sup>H</sup> einH<sup>ω</sup>

� �

Recent Advances in Integral Equations

<sup>F</sup> <sup>1</sup>

$$\begin{split} I &:= \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \frac{g(t\_1, t\_2) e^{it\_1 \alpha\_1 + it\_2 \alpha\_2}}{\left(1 + 2\pi a + 2\pi a t\_1^2\right) \left(1 + 2\pi a + 2\pi a t\_2^2\right)} \\ &= \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \frac{e^{it\_1 \alpha\_1 + it\_2 \alpha\_2}}{\left(1 + 2\pi a + 2\pi a t\_1^2\right) \left(1 + 2\pi a + 2\pi a t\_2^2\right)} \\ & \cdot \sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} g(n\_1 H\_1, n\_2 H\_2) \frac{\sin \mathcal{Q}\_1(t\_1 - n\_1 H\_1)}{\mathcal{Q}\_1(t\_1 - n\_1 H\_1)} \frac{\sin \mathcal{Q}\_2(t\_2 - n\_2 H\_2)}{\mathcal{Q}\_2(t\_2 - n\_2 H\_2)} dt\_1 dt\_2 \\ &= \sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} g(n\_1 H\_1, n\_2 H\_2) \int\_{-\infty}^{\infty} \frac{1}{1 + 2\pi a + 2\pi a t\_1^2} \frac{\sin \mathcal{Q}\_1(t\_1 - n\_1 H\_1)}{\mathcal{Q}\_1(t\_1 - n\_1 H\_1)} e^{it\_1 \alpha\_1} dt\_1 \\ & \cdot \int\_{-\infty}^{\infty} \frac{1}{1 + 2\pi a + 2\pi a t\_2^2} \frac{\sin \mathcal{Q}\_2(t\_2 - n\_2 H\_2)}{\mathcal{Q}\_2(t\_2 - n\_2 H\_2)} e^{it\_1 \alpha\_1} dt\_2 \end{split}$$

By Lemma 3.1 and the FOIL method, Eq. (10) is true.

Lemma 3.3 For each arbitrarily small c>0 and ω ∈½ � �Ω þ c; Ω � c ,

$$\sum\_{n=-\infty}^{\infty} \left| \frac{e^{a(\alpha - \mathcal{Q})}}{a - inH} + \frac{e^{-a(\alpha + \mathcal{Q})}}{a + inH} \right|^2 = O\left(\frac{e^{-2ac}}{a}\right). \tag{11}$$

Proof. By the inequality ∣a þ b∣ <sup>2</sup> ≤2 ∣a∣ <sup>2</sup> <sup>þ</sup> <sup>∣</sup>b<sup>∣</sup> <sup>2</sup> � �,

$$\sum\_{n=-\infty}^{\infty} \left| \frac{e^{a(a-\varOmega)}}{a-inH} + \frac{e^{-a(a+\varOmega)}}{a+inH} \right|^2 \le 2 \sum\_{n=-\infty}^{\infty} \left[ \left| \frac{e^{a(a-\varOmega)}}{a-inH} \right|^2 + \left| \frac{e^{-a(a+\varOmega)}}{a+inH} \right|^2 \right]$$

$$\le 4 \sum\_{n=-\infty}^{\infty} \frac{e^{-2ac}}{a^2 + (nH)^2} \le \frac{4}{H} e^{-2ac} \int\_{-\infty}^{\infty} \frac{dx}{a^2 + x^2} + \frac{4}{a^2} e^{-2ac} = \frac{4\pi e^{-2ac}}{Ha} + \frac{4}{a^2} e^{-2ac}.$$

Lemma 3.4 For each arbitrarily small c>0 and ð Þ ω1; ω<sup>2</sup> ∈½�Ω<sup>1</sup> þ c; Ω<sup>1</sup> � c�� ½ � �Ω<sup>2</sup> þ c; Ω<sup>2</sup> � c ,

$$\begin{split} \sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} \mathcal{g}(n\_1 H\_1, n\_2 H\_2) \frac{(-1)^{n\_1 + n\_2}}{(4\pi a\alpha)^2} \left[ \frac{e^{a(\boldsymbol{w}\_1 - \Omega\_1)}}{a - in\_1 H\_1} + \frac{e^{-a(\boldsymbol{w}\_1 + \Omega\_1)}}{a + in\_1 H\_1} \right] \\ \left[ \frac{e^{a(\boldsymbol{w}\_2 - \Omega\_2)}}{a - in\_2 H\_2} + \frac{e^{-a(\boldsymbol{w}\_2 + \Omega\_2)}}{a + in\_2 H\_2} \right], \quad = O(ae^{-2ac}). \end{split} \tag{12}$$

for α ! þ0 and g that is Ω-band-limited.

Proof. By the Cauchy inequality,

$$\begin{split} \left| \sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} g(n\_1 H\_1, n\_2 H\_2) \left[ \frac{e^{a(\alpha\_1 - \Omega\_1)}}{a - in\_1 H\_1} + \frac{e^{-a(\alpha\_1 + \Omega\_1)}}{a + in\_1 H\_1} \right] \left[ \frac{e^{a(\alpha\_2 - \Omega\_2)}}{a - in\_2 H\_2} + \frac{e^{-a(\alpha\_2 + \Omega\_2)}}{a + in\_2 H\_2} \right] \right|^2 \\ \leq \sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} \left| g(n\_1 H\_1, n\_2 H\_2) \right|^2 \\ \sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} \left| \left[ \frac{e^{a(\alpha\_1 - \Omega\_1)}}{a - in\_1 H\_1} + \frac{e^{-a(\alpha\_1 + \Omega\_1)}}{a + in\_1 H\_1} \right] \left[ \frac{e^{a(\alpha\_2 - \Omega\_2)}}{a - in\_2 H\_2} + \frac{e^{-a(\alpha\_2 + \Omega\_2)}}{a + in\_2 H\_2} \right] \right|^2, \end{split}$$

$$\sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} \left| \left[ \frac{e^{a(\mathbf{u}\_1 - \Omega\_1)}}{a - in\_1 H\_1} + \frac{e^{-a(\mathbf{u}\_1 + \Omega\_1)}}{a + in\_1 H\_1} \right] \left[ \frac{e^{a(\mathbf{u}\_2 - \Omega\_2)}}{a - in\_2 H\_2} + \frac{e^{-a(\mathbf{u}\_2 + \Omega\_2)}}{a + in\_2 H\_2} \right] \right|^2$$

$$= \sum\_{n\_1 = -\infty}^{\infty} \left| \left[ \frac{e^{a(\mathbf{u}\_1 - \Omega\_1)}}{a - in\_1 H\_1} + \frac{e^{-a(\mathbf{u}\_1 + \Omega\_1)}}{a + in\_1 H\_1} \right] \right|^2 \sum\_{n\_2 = -\infty}^{\infty} \left| \left[ \frac{e^{a(\mathbf{u}\_1 - \Omega\_1)}}{a - in\_1 H\_1} + \frac{e^{-a(\mathbf{u}\_1 + \Omega\_1)}}{a + in\_1 H\_1} \right] \right|^2 \dots$$

$$\sum\_{n=-\infty}^{\infty} \left| \frac{1}{1 + 2\pi a + 2\pi a (nH)^2} \right|^2 = O\left(\frac{1}{\sqrt{a}}\right). \tag{13}$$

$$\stackrel{\circ}{\nabla} \quad \left| \begin{array}{c} \mathbf{1} \\ \hline \\ \end{array} \right|^2 \prec \left| \begin{array}{c} \mathbf{1} \\ \hline \\ \end{array} \right|^2 + \left| \begin{array}{c} \mathbf{1} \\ \hline \\ \end{array} \right|^2 + \left| \begin{array}{c} \mathbf{1} \\ \\ \end{array} \right|^2.$$

$$\sum\_{n=-\infty}^{\infty} \left| \frac{1}{1 + 2\pi a + 2\pi a (nH)^2} \right| \le \left| \frac{1}{1 + 2\pi a} \right|^2 + \sum\_{n \ne 0} \left| \frac{1}{1 + 2\pi a + 2\pi a (nH)^2} \right|^2$$

$$\sum\_{n\neq 0} \left| \frac{1}{1 + 2\pi a + 2\pi a(nH)^2} \right|^2 \le 2 \sum\_{n=1}^{\infty} \frac{1}{\left[1 + 2\pi a + 2\pi a(nH)^2\right]^2} \le 1$$

$$\frac{2}{H} \int\_0^{\infty} \frac{dx}{\left(1 + 2\pi a + 2\pi a x^2\right)^2} = O\left(\frac{1}{\sqrt{a}}\right).$$

$$\sum\_{n=-\infty}^{\infty} \left| \frac{1}{1 + 2\pi\alpha + 2\pi\alpha(nH)^2} \right|^2 = O\left(\frac{1}{\sqrt{a}}\right).$$

$$\sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} g(n\_1 H\_1, n\_2 H\_2) \left[ \frac{e^{in\_1 H\_1 a\_1}}{1 + 2\pi a + 2\pi a (n\_1 H\_1)^2} \frac{(-1)^{n\_2}}{4\pi a a} \left( \frac{e^{a(a\_2 - \mathcal{Q}\_2)}}{a - in\_2 H\_2} + \frac{e^{-a(a\_2 + \mathcal{Q}\_2)}}{a + in\_2 H\_2} \right) \right]$$
 
$$= O\left(a^2 e^{-ac}\right),$$

$$\begin{split} \left| \sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} \operatorname{g}(n\_1 H\_1, n\_2 H\_2) \left[ \frac{e^{in\_1 H\_1 \alpha\_1}}{1 + 2\pi a + 2\pi a (n\_1 H\_1)^2} \right] \left[ \frac{e^{a(\alpha\_2 - \Omega\_2)}}{a - in\_2 H\_2} + \frac{e^{-a(\alpha\_2 + \Omega\_2)}}{a + in\_2 H\_2} \right] \right|^2 \\ \leq \sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} \left| \operatorname{g}(n\_1 H\_1, n\_2 H\_2) \right|^2 \\\\ \sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} \left| \left[ \frac{e^{in\_1 H\_1 \alpha\_1}}{1 + 2\pi a + 2\pi a (n\_1 H\_1)^2} \right] \left[ \frac{e^{a(\alpha\_2 - \Omega\_2)}}{a - in\_2 H\_2} + \frac{e^{-a(\alpha\_2 + \Omega\_2)}}{a + in\_2 H\_2} \right] \right|^2, \end{split}$$

$$\sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} \left| \left[ \frac{e^{in\_1 H\_1 \omega\_1}}{1 + 2\pi\alpha + 2\pi\alpha (n\_1 H\_1)^2} \right] \left[ \frac{e^{a(\omega\_2 - \Omega\_2)}}{a - in\_2 H\_2} + \frac{e^{-a(\omega\_2 + \Omega\_2)}}{a + in\_2 H\_2} \right] \right|^2$$

$$= \sum\_{n\_1 = -\infty}^{\infty} \left| \frac{e^{in\_1 H\_1 \omega\_1}}{1 + 2\pi\alpha + 2\pi\alpha (n\_1 H\_1)^2} \right|^2 \sum\_{n\_2 = -\infty}^{\infty} \left| \left[ \frac{e^{a(\omega\_2 - \Omega\_2)}}{a - in\_2 H\_2} + \frac{e^{-a(\omega\_2 + \Omega\_2)}}{a + in\_2 H\_2} \right] \right|^2$$

$$\left| \sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} \frac{\eta(n\_1 H\_1, n\_2 H\_2)}{\left| \left[ \mathbf{1} + 2\pi a + 2\pi a (n\_1 H\_1)^2 \right] \left[ \mathbf{1} + 2\pi a + 2\pi a (n\_2 H\_2)^2 \right] \right|} \right| = O\left(\frac{\delta}{a}\right) \tag{15}$$

$$\sum\_{n=-\infty}^{\infty} \left| \frac{1}{1 + 2\pi\alpha + 2\pi\alpha(nH\_1)^2} \right| \le \left| \frac{1}{1 + 2\pi\alpha} \right| + \sum\_{n \ne 0} \left| \frac{1}{1 + 2\pi\alpha + 2\pi\alpha(nH\_1)^2} \right|,$$

$$\begin{aligned} \sum\_{n \neq 0} \left| \frac{1}{1 + 2\pi a + 2\pi a (nH\_1)^2} \right| &\le 2 \sum\_{n=1}^{\infty} \frac{1}{1 + 2\pi a + 2\pi a (nH\_1)^2} \\ &\le \frac{2}{H\_1} \int\_0^{\infty} \frac{dx}{1 + 2\pi a + 2\pi a x^2} = O\left(\frac{1}{\sqrt{a}}\right). \end{aligned}$$

$$\sum\_{n=-\infty}^{\infty} \left| \frac{1}{1 + 2\pi a + 2\pi a (nH\_1)^2} \right| = O\left( 1/\sqrt{a} \right).$$

$$\sum\_{n=-\infty}^{\infty} \left| \frac{1}{1 + 2\pi\alpha + 2\pi\alpha (nH\_2)^2} \right| = O\left(\frac{1}{\sqrt{a}}\right).$$

$$\begin{split} &H\_{1}H\_{2} \sum\_{n\_{1}=-\infty}^{\infty} \sum\_{n\_{2}=-\infty}^{\infty} \frac{f\_{\,^{T}}(n\_{1}H\_{1},n\_{2}H\_{2})}{[1+2\pi a + 2\pi a(n\_{1}H\_{1})^{2}][1+2\pi a + 2\pi a(n\_{2}H\_{2})^{2}]}e^{i n\_{1}H\_{1}a\_{1} + i n\_{2}H\_{2}a\_{2}}, \\ &= \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \frac{f\_{\,^{T}}(t\_{1},t\_{2})e^{it\_{1}a\_{1} + it\_{2}a\_{2}}dt\_{1}dt\_{2}}{\left(1+2\pi a + 2\pi a t\_{1}^{2}\right)\left(1+2\pi a + 2\pi at\_{2}^{2}\right)} + O\left(a^{\frac{1}{2}}e^{-at}\right). \end{split}$$

### 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 Þ¼ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> <sup>H</sup>1H<sup>2</sup> <sup>∑</sup> ∞ n1¼�∞ ∑ ∞ n2¼�∞ ½ � fTð Þþ n1H1; n2H2 <sup>η</sup>ð Þ n1H1; n2H2 <sup>e</sup>in1H1ω1þin2H2ω<sup>2</sup> <sup>½</sup><sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα n1H1Þ<sup>2</sup> ð �½<sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα n2H2Þ<sup>2</sup> ð � <sup>P</sup>Ωð 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> ¼ H1H<sup>2</sup> ∑ ∞ n1¼�∞ ∑ ∞ n2¼�∞ <sup>f</sup> <sup>T</sup>ð Þ <sup>n</sup>1H1; <sup>n</sup>2H<sup>2</sup> ein1H1ω1þin2H2ω<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> ½ � <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> ½ � <sup>P</sup>Ωð 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> þH1H<sup>2</sup> ∑ ∞ n1¼�∞ ∑ ∞ n2¼�∞ ηð Þ n1H1; n2H<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> ½ � <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> ½ �<sup>e</sup> in1H1ω1þin2H2ω2PΩð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> ¼ ∞ð �∞ ∞ð �∞ <sup>f</sup> <sup>T</sup>ð Þ <sup>t</sup>1; <sup>t</sup><sup>2</sup> eit1ω1þit2ω2dt1dt<sup>2</sup> 1 þ 2πα þ 2παt 2 1 � � <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα<sup>t</sup> 2 2 � � � ∞ð �∞ ∞ð �∞ f <sup>T</sup>ð Þ t1; t<sup>2</sup> e it1ω1þit2ω2dt1dt<sup>2</sup> 2 4 3 5PΩð Þ ω1; ω<sup>2</sup> þH1H<sup>2</sup> ∑ ∞ n1¼�∞ ∑ ∞ n2¼�∞ <sup>η</sup>ð Þ <sup>n</sup>1H1; <sup>n</sup>2H<sup>2</sup> ein1H1ω1þin2H2ω<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 <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>P</sup>Ωð Þþ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> O a<sup>1</sup> 2e �ac � �:

ðð

� � � � � � �

≤

� � � � � � � 4πα þ 2παt

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

f <sup>T</sup>ð Þ t1; t<sup>2</sup> � � �

4πα þ 2παt

2 <sup>1</sup> þ 2παt 2

1 þ 2πα þ 2παt

analysis of the regularized Fourier series according to the L<sup>2</sup>

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

� �ein1H1ω1þin2H2ω<sup>2</sup>

By Lemma 3.5, we have next lemma.

½ �� � �Ω1; Ω<sup>1</sup> ½ � Ω2; Ω<sup>2</sup> as δ ! 0.

2 <sup>1</sup> þ 2παt 2

1 þ 2πα þ 2παt

�dt1dt<sup>2</sup> < ε

<sup>2</sup> þ 2πα þ 2παt

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

<sup>2</sup> þ 2πα þ 2παt

! 0

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

tions <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

ηð Þ n1H1; n2H<sup>2</sup> 1 þ 2πα þ 2παð Þ n1H<sup>1</sup> <sup>2</sup> ½ � 1 þ 2πα þ 2παð Þ n2H<sup>2</sup> <sup>2</sup> ½ �

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

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

<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

<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 <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>P</sup>Ωð 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>4</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>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> � � <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 <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

2 1 � � <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα<sup>t</sup>

2 1 � � <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα<sup>t</sup>

2 1 � � <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα<sup>t</sup>

2 1 � � <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα<sup>t</sup>

2 2

> 2 2

> > � � � �

2

� �

2 2 � � <sup>f</sup> <sup>T</sup>ð Þ <sup>t</sup>1; <sup>t</sup><sup>2</sup> <sup>e</sup>

it1ω1þit2ω2dt1dt<sup>2</sup>

it1ω1þit2ω2dt1dt<sup>2</sup>


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

α � � � � � � � � �

� � � � � � �

� �

2 2 � � <sup>f</sup> <sup>T</sup>ð Þ <sup>t</sup>1; <sup>t</sup><sup>2</sup> <sup>e</sup>

∣t1∣≥M or∣t2∣≥M

and

ðð

∣t1∣≥M or∣t2∣≥M

ðð

∣t1∣ ≤ M and ∣t2∣ ≤ M

as α ! 0.

4. Error analysis

presented.

∑ ∞ n1¼�∞

L2

Proof.

¼ H1H<sup>2</sup> ∑ ∞ n1¼�∞

¼ �H1H<sup>2</sup> ∑

53

^<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>

∞ n1¼�∞

∑ ∞ n2¼�∞

> ∑ ∞ n2¼�∞

Lemma 4.1

Section 1.

∑ ∞ n2¼�∞

that α δð Þ! 0 and <sup>δ</sup><sup>2</sup>

� � � �

This implies

$$\begin{split} & \left| \hat{f}\_{a}(\mathbf{u}\_{1},\mathbf{u}\_{2}) - \hat{f}\_{T}(\mathbf{u}\_{1},\mathbf{u}\_{2}) \right| \\ & \leq \left| \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \frac{4\pi a + 2\pi a t\_{1}^{2} + 2\pi a t\_{2}^{2} + \left( 2\pi a + 2\pi a t\_{1}^{2} \right) \left( 2\pi a + 2\pi a t\_{2}^{2} \right)}{\left( 1 + 2\pi a + 2\pi a t\_{1}^{2} \right) \left( 1 + 2\pi a + 2\pi a t\_{2}^{2} \right)} f\_{T}(t\_{1},t\_{2}) e^{i t\_{1} \alpha\_{1} + i t\_{2} \alpha\_{2}} dt\_{1} dt\_{2} \right| \\ & + H\_{1} H\_{2} \left| \sum\_{n\_{1} = -\infty}^{\infty} \sum\_{n\_{2} = -\infty}^{\infty} \frac{\eta(n\_{1} H\_{1}, n\_{2} H\_{2})}{[1 + 2\pi a + 2\pi a (n\_{1} H\_{1})^{2}][1 + 2\pi a + 2\pi a (n\_{2} H\_{2})^{2}]} e^{i n\_{1} H\_{1} \alpha\_{1} + i n\_{2} H\_{2} \alpha\_{2}} \right| \\ & + O\left( a^{\frac{1}{2} - a^{c}} \right) \end{split}$$

where

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

For any ε>0, there exists M>0 such that

$$\int \int\_{|t\_1| \ge M \text{ or } |t\_2| \ge M} |f\_{T\_1}(t\_1, t\_2)| dt\_1 dt\_2 < \varepsilon.$$

Then

$$\begin{split} & \left| \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \frac{4\pi a + 2\pi a t\_1^2 + 2\pi a t\_2^2 + \left( 2\pi a + 2\pi a t\_1^2 \right) \left( 2\pi a + 2\pi a t\_2^2 \right)}{\left( 1 + 2\pi a + 2\pi a t\_1^2 \right) \left( 1 + 2\pi a + 2\pi a t\_2^2 \right)} f\_T(t\_1, t\_2) e^{\bar{t}\_1 w\_1 + \bar{t}\_2 w\_2} dt\_1 dt\_2 \right| \\ & \leq \left| \iint\_{\substack{|t\_1| \leq M \ m \, |t\_2| \leq M}} \frac{4\pi a + 2\pi a t\_1^2 + 2\pi a t\_2^2 + \left( 2\pi a + 2\pi a t\_1^2 \right) \left( 2\pi a + 2\pi a t\_2^2 \right)}{\left( 1 + 2\pi a + 2\pi a t\_1^2 \right) \left( 1 + 2\pi a + 2\pi a t\_2^2 \right)} f\_T(t\_1, t\_2) e^{\bar{t}\_1 w\_1 + \bar{t}\_2 w\_2} dt\_1 dt\_2 \right| \\ & \quad + \left| \iint\_{\substack{|t\_1| \leq M \ l \, |t\_2| \geq M}} \frac{4\pi a + 2\pi a t\_1^2 + 2\pi a t\_2^2 + \left( 2\pi a + 2\pi a t\_1^2 \right) \left( 2\pi a + 2\pi a t\_2^2 \right)}{\left( 1 + 2\pi a + 2\pi a t\_1^2 \right) \left( 1 + 2\pi a + 2\pi a t\_2^2 \right)} f\_T(t\_1, t\_2) e^{\bar{t}\_1 w\_1 + \bar{t}\_2 w\_2} dt\_1 dt\_2 \right|. \end{split}$$

where

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

$$\begin{aligned} & \left| \iint\_{|t\_1| \ge M \text{ or } |t\_2| \ge M} \frac{4\pi a + 2\pi a\_1^2 + 2\pi a\_2^2 + \left( 2\pi a + 2\pi a\_1^2 \right) \left( 2\pi a + 2\pi a\_2^2 \right)}{\left( 1 + 2\pi a + 2\pi a\_1^2 \right) \left( 1 + 2\pi a + 2\pi a\_2^2 \right)} f\_T(t\_1, t\_2) e^{i t\_1 a\_1 + i t\_2 a\_2} d t\_1 d t\_2 \right| \\ & \le \iint\_{|t\_1| \ge M \text{ or } |t\_2| \ge M} \left| f\_T(t\_1, t\_2) \right| d t\_1 d t\_2 < \varepsilon \\ & \quad \text{and} \\ & \left| \iint\_{|t\_1| \le M \text{ and } |t\_2| \le M} \frac{4\pi a + 2\pi a\_1^2 + 2\pi a\_2^2 + \left( 2\pi a + 2\pi a\_1^2 \right) \left( 2\pi a + 2\pi a\_1^2 \right)}{\left( 1 + 2\pi a + 2\pi a\_1^2 \right) \left( 1 + 2\pi a + 2\pi a\_2^2 \right)} f\_T(t\_1, t\_2) e^{i t\_1 a\_1 + i t\_2 a\_2} d t\_1 d t\_2 \right| \\ & \qquad \qquad \qquad \to 0 \end{aligned}$$

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 Þ¼ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> <sup>H</sup>1H<sup>2</sup> <sup>∑</sup>

Recent Advances in Integral Equations

¼ H1H<sup>2</sup> ∑ ∞ n1¼�∞

þH1H<sup>2</sup> ∑ ∞ n1¼�∞

�∞

þH1H<sup>2</sup> ∑ ∞ n1¼�∞

^<sup>f</sup> <sup>α</sup>ð Þ� <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> ^<sup>f</sup> <sup>T</sup>ðω1; <sup>ω</sup>2<sup>Þ</sup>

4πα þ 2παt

∑ ∞ n2¼�∞

∞ð

�∞

¼ ∞ð

This implies

ð<sup>∞</sup> �∞

> ∞ n1¼�∞

þH1H<sup>2</sup> ∑

where

Then

ð<sup>∞</sup> �∞

ðð

∣t1∣ ≤ M and ∣t2∣ ≤ M

ðð

∣t1∣≥M or∣t2∣≥M

where

4πα þ 2παt

2 <sup>1</sup> þ 2παt 2

ð<sup>∞</sup> �∞

> � � � � � � �

≤

þ

52

� � � � � � �

� � � � �

þO a<sup>1</sup>

∑ ∞ n1¼�∞

� � � � � � � � � � �

<sup>2</sup>e�ac � �

> ∑ ∞ n2¼�∞

� � �

> ≤ ð<sup>∞</sup> �∞

� � � � � 2 4

∞ n1¼�∞

∑ ∞ n2¼�∞

∑ ∞ n2¼�∞

∑ ∞ n2¼�∞

1 þ 2πα þ 2παt

� � �

2 <sup>1</sup> þ 2παt 2

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

For any ε>0, there exists M>0 such that

1 þ 2πα þ 2παt

2 <sup>1</sup> þ 2παt 2

2 <sup>1</sup> þ 2παt 2

1 þ 2πα þ 2παt

4πα þ 2παt

4πα þ 2παt

ð ð

1 þ 2πα þ 2παt

∑ ∞ n2¼�∞

<sup>f</sup> <sup>T</sup>ð Þ <sup>t</sup>1; <sup>t</sup><sup>2</sup> eit1ω1þit2ω2dt1dt<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

2 1 � � <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα<sup>t</sup>

½ � fTð Þþ n1H1; n2H2 <sup>η</sup>ð Þ n1H1; n2H2 <sup>e</sup>in1H1ω1þin2H2ω<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> ½ � <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> ½ � <sup>P</sup>Ωð 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>f</sup> <sup>T</sup>ð Þ <sup>n</sup>1H1; <sup>n</sup>2H<sup>2</sup> ein1H1ω1þin2H2ω<sup>2</sup>

ηð Þ n1H1; n2H<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> ½ � <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> ½ �<sup>e</sup>

> 2 2

<sup>η</sup>ð Þ <sup>n</sup>1H1; <sup>n</sup>2H<sup>2</sup> ein1H1ω1þin2H2ω<sup>2</sup>

<sup>2</sup> þ 2πα þ 2παt

ηð Þ n1H1; n2H<sup>2</sup> <sup>½</sup><sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα <sup>n</sup>1H1Þ<sup>2</sup> ð �½<sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα <sup>n</sup>2H2Þ<sup>2</sup> ð �<sup>e</sup>

2 1 � � <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα<sup>t</sup>

ηð Þ n1H1; n2H<sup>2</sup>

∣t1∣≥M or∣t2∣≥M

<sup>2</sup> þ 2πα þ 2παt

2 1 � � <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα<sup>t</sup>

1 þ 2πα þ 2παt

∞ð

∞ð

f <sup>T</sup>ð Þ t1; t<sup>2</sup> e

<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>P</sup>Ωð Þþ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> O a<sup>1</sup>

� �

2 2 � � <sup>f</sup> <sup>T</sup>ð Þ <sup>t</sup>1; <sup>t</sup><sup>2</sup> <sup>e</sup>

�∞

�∞

2 1 � � <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα<sup>t</sup>

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

�dt1dt<sup>2</sup> < ε:

2 2

2 2 � � <sup>f</sup> <sup>T</sup>ð Þ <sup>t</sup>1; <sup>t</sup><sup>2</sup> <sup>e</sup>

� �

2 2 � � <sup>f</sup> <sup>T</sup>ð Þ <sup>t</sup>1; <sup>t</sup><sup>2</sup> <sup>e</sup>

� �

2 2 2 2

� �

2 1 � � <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα<sup>t</sup>

2 2 � � <sup>f</sup> <sup>T</sup>ð Þ <sup>t</sup>1; <sup>t</sup><sup>2</sup> <sup>e</sup>

2 1 � � <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα<sup>t</sup>

f <sup>T</sup>ð Þ t1; t<sup>2</sup> � � �

2 1 � � <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα<sup>t</sup>

<sup>2</sup> þ 2πα þ 2παt

2 1 � � <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα<sup>t</sup>

<sup>2</sup> þ 2πα þ 2παt

2 1 � � <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα<sup>t</sup>

� � �

<sup>½</sup><sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα n1H1Þ<sup>2</sup> ð �½<sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>πα n2H2Þ<sup>2</sup> ð � <sup>P</sup>Ωð 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>

it1ω1þit2ω2dt1dt<sup>2</sup>

2 2

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

3 5PΩð Þ ω1; ω<sup>2</sup>

> 2e �ac � � :

in1H1ω1þin2H2ω<sup>2</sup>

� � � � � �

in1H1ω1þin2H2ω<sup>2</sup>

it1ω1þit2ω2dt1dt<sup>2</sup>

� � � � �

it1ω1þit2ω2dt1dt<sup>2</sup>

it1ω1þit2ω2dt1dt<sup>2</sup>

� � � � � � �

� � � � � � � ,

it1ω1þit2ω2dt1dt<sup>2</sup>

� � � � �

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

� � � � �
