2. The ill-posedness

We will first study the ill-posedness of the problem (3) in the noisy case (4). The concept of ill-posed problems was introduced in [15]. Here we borrow the following definition from it.

Definition 2.1 Assume A: D ! U is an operator in which D and U are metric spaces with distances ρDð Þ ∗; ∗ and ρUð Þ ∗; ∗ , respectively. The problem

$$Az = u.\tag{6}$$

of determining a solution z in the space D from the "initial data" u in the space U is said to be well-posed on the pair of metric spaces ð Þ D; U in the sense of Hadamard if the following three conditions are satisfied:


$$
\rho\_U(\mu\_1, \mu\_2) < \delta \Rightarrow \rho\_D(z\_1, z\_2) < \in \square
$$

In other words, the inverse mapping A�<sup>1</sup> is uniformly continuous. Problems that violate any of the three conditions are said to be ill-posed.

In this section, we discuss the ill-posedness of <sup>A</sup>^<sup>f</sup> <sup>¼</sup> <sup>f</sup> on the pair of Banach spaces (L<sup>2</sup> ½ �� � �Ω1; <sup>Ω</sup><sup>1</sup> ½ � <sup>Ω</sup>2; <sup>Ω</sup><sup>2</sup> , l<sup>∞</sup>(Z<sup>2</sup> )), where ^fð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> is given by the Fourier series in Eq. (3).

The operator A in Eq. (6) is defined by the following formula:

$$A\hat{f} = f,\tag{7}$$

where ¼ f g f nð Þ <sup>1</sup>H1; n2H<sup>2</sup> : n1∊Z; n2∊Z .

As usual, l <sup>∞</sup> is the space a nð Þ: <sup>n</sup>∊Z<sup>2</sup> � � of bounded sequences. The norm of <sup>l</sup> <sup>∞</sup> is defined by

$$||\mathfrak{a}||\_{l^{\infty}} = \sup\_{\mathfrak{n} \in \mathbb{Z}^2} |\mathfrak{a}(\mathfrak{n})|,$$

where

For band-limited signals, we have the following sampling theorem [4, 10, 11].

Calculating the Fourier transform of f tð Þ <sup>1</sup>; t<sup>2</sup> by the formula (2), we have the

f nð Þ <sup>1</sup>H1; n2H<sup>2</sup> e

where PΩð Þ¼ ω1; ω<sup>2</sup> 1½ �� � �Ω1;Ω<sup>1</sup> ½ � <sup>Ω</sup>2;Ω<sup>2</sup> (ω1, ω2) is the characteristic function of

The noise in the two-dimensional case is discussed in [5, 6], and the Tikhonov regularization method is used. However, there is too much computation in the Tikhonov regularization method since the solution of an Euler equation is required. The ill-posedness in the one-dimensional case is considered in [12, 13]. The

> f nH ð ÞeinH<sup>ω</sup> 1 þ 2πα þ 2παð Þ n1H<sup>1</sup>

> > f tð Þeiω<sup>t</sup> dt

in [14]. The regularized Fourier transform was found by finding the minimizer

In this chapter, we will find a reliable algorithm for this ill-posed problem using

We will first study the ill-posedness of the problem (3) in the noisy case (4). The concept of ill-posed problems was introduced in [15]. Here we borrow the following

a two-dimensional regularized Fourier series. In Section 2, the ill-posedness is discussed in the two-dimensional case. In Section 3, the regularized Fourier series and the proof of the convergence property are given. The bias and variance of regularized Fourier series are given in Section 4. The algorithm and the experimental results of numerical examples are given in Section 5. Finally, the conclusion is

1 þ 2πα þ 2παt2

In many practical problems, the samples f g f nð Þ <sup>1</sup>H1; n2H<sup>2</sup> are noisy:

sin Ω1ð Þ t<sup>1</sup> � n1H<sup>1</sup> Ω1ð Þ t<sup>1</sup> � n1H<sup>1</sup>

f nð Þ¼ <sup>1</sup>H1; n2H<sup>2</sup> f <sup>T</sup>ð Þþ n1H1; n2H<sup>2</sup> ηð Þ n1H1; n2H<sup>2</sup> , (4)

∣ηð Þ n1H1; n2H<sup>2</sup> ∣ ≤ δ, (5)

<sup>2</sup> PΩð Þ ω

sin Ω2ð Þ t<sup>2</sup> � n2H<sup>2</sup> Ω2ð Þ t<sup>2</sup> � n2H<sup>2</sup>

in1H1ω1þin2H2ω2PΩð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> , (3)

, (2)

For the two-dimensional band-limited function above, we have

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

∑ ∞ n2¼�∞

f tð Þ¼ <sup>1</sup>; t<sup>2</sup> ∑

∞ n1¼�∞

Recent Advances in Integral Equations

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

½ �� � �Ω1; Ω<sup>1</sup> ½ � Ω2; Ω<sup>2</sup> .

regularized Fourier series

given in Section 6.

definition from it.

46

2. The ill-posedness

∑ ∞ n2¼�∞

formula which is same as the Fourier series

where f g ηð Þ n1H1; n2H<sup>2</sup> is the noise

of the Tikhonov's smoothing functional.

and f <sup>T</sup> ∈L<sup>2</sup> is the exact band-limited signal.

^<sup>f</sup> <sup>α</sup>ð Þ¼ <sup>ω</sup> <sup>H</sup> <sup>∑</sup>

∞ n¼�∞

in [12] is given based on the regularized Fourier transform

ð<sup>∞</sup> �∞

Fα½ �¼ f

∞ n1¼�∞

where H<sup>1</sup> ≔ π=Ω<sup>1</sup> and H<sup>2</sup> ≔ π=Ω2.

