Approximation of Bivariate Functions via Smooth Extensions

For a smooth bivariate function defined on a general domain with arbitrary shape, it is difficult to do Fourier approximation or wavelet approximation. In order to solve these problems, in this paper, we give an extension of the bivariate function on a general domain with arbitrary shape to a smooth, periodic function in the whole space or to a smooth, compactly supported function in the whole space. These smooth extensions have simple and clear representations which are determined by this bivariate function and some polynomials. After that, we expand the smooth, periodic function into a Fourier series or a periodic wavelet series or we expand the smooth, compactly supported function into a wavelet series. Since our extensions are smooth, the obtained Fourier coefficients or wavelet coefficients decay very fast. Since our extension tools are polynomials, the moment theorem shows that a lot of wavelet coefficients vanish. From this, with the help of well-known approximation theorems, using our extension methods, the Fourier approximation and the wavelet approximation of the bivariate function on the general domain with small error are obtained.


Introduction
In the recent several decades, various approximation tools have been widely developed [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. For example, a smooth periodic function can be approximated by trigonometric polynomials; a square-integrable smooth function can be expanded into a wavelet series and be approximated by partial sum of the wavelet series; and a smooth function on a cube can be approximated well by polynomials. However, for a smooth function on a general domain with arbitrary shape, even if it is infinitely many time differentiable, it is difficult to do Fourier approximation or wavelet approximation. In this paper, we will extend a function on general domain with arbitrary shape to a smooth, periodic function in the whole space or to a smooth, compactly supported function in the whole space. After that, it will be easy to do Fourier approximation or wavelet approximation. For the higherdimensional case, the method of smooth extensions is similar to that in the two-dimensional case, but the representations of smooth extensions will be too complicated. Therefore, in this paper, we mainly consider the smooth extension of a bivariate function on a planar domain. By the way, for the one-dimensional case, since the bounded domain is reduced to a closed interval, the smooth extension can be regarded as a corollary of the two-dimensional case.
This paper is organized as follows. In Section 2, we state the main theorems on the smooth extension of the function on the general domain and their applications. In Sections 3 and 4, we give a general method of smooth extensions and complete the proofs of the main theorems. In Section 5, we use our extension method to discuss two important special cases of smooth extensions.
Throughout this paper, we denote = [0, 1] 2 and the interior of by and always assume that Ω is a simply connected domain. We say that ∈ (Ω) if the derivatives ( + / ) are continuous on Ω for 0 ≤ + ≤ . We say that ∈ ∞ (Ω) if all derivatives ( + / ) are continuous on Ω for , ∈ Z + . We say that a function ℎ( , ) is a -periodic function if ℎ( + , + ) = ℎ( , ) (( , ) ∈ 2 The Scientific World Journal ; , ∈ Z), where is an integer. We appoint that 0! = 1 and the notation [ ] is the integral part of the real number .

Main Theorems and Applications
In this section, we state the main results of smooth extensions and their applications in Fourier analysis and wavelet analysis.

Main Theorems.
Our main theorems are stated as follows.
where Ω ⊂ and the boundary Ω is a piecewise infinitely many time smooth curve. Then for any ∈ Z + there is a function ∈ ( ) such that where is a positive integer and each coefficient is constant.
In Sections 3 and 4, we give constructive proofs of Theorems 1-3. In these three theorems, we assume that ∈ ∞ (Ω). If ∈ (Ω) ( is a nonnegative integer), by using the similar method of arguments of Theorems 1-3, we also can obtain the corresponding results.

Applications.
Here we show some applications of these theorems.

Approximation by Polynomials.
Let be the smooth extension of from Ω to which is stated as in Theorem 1. Then ∈ ( ) and = on Ω. By Δ , denote the set of all bivariate polynomials in the form ∑ where ‖ ⋅ ‖ ( ) is the norm of the space ( ). The righthand side of formula (2) is the best approximation of the extension in Δ . By (2), we know that the approximation problem of by polynomials on a domain Ω is reduced to the well-known approximation problem of its smooth extension by polynomials on the square [4, 10].

Fourier Analysis
(i) Approximation by Trigonometric Polynomials. Let be the smooth periodic extension of as in Theorem 2. Then ∈ (R 2 ) and = on Ω. By the well-known results [5,10], we know that the smooth periodic function can be approximated by bivariate trigonometric polynomials very well. Its approximation error can be estimated by the modulus of continuity of its time derivatives.
By Δ * , denote the set of all bivariate trigonometric polynomials in the form By Theorem 2, we have From this and Theorem 2, we see that the approximation problem of on Ω by trigonometric polynomials is reduced to a well-known approximation problem of smooth periodic functions [5,7,10].
(ii) Fourier Series. We expand into a Fourier series [9] ( , ) = ∑ . By Theorem 2, we obtain that, for ( , ) ∈ Ω, Denote the partial sum Then we have Since the smooth periodic function can be approximated well by the partial sum of its Fourier series [5,7,10], from this inequality, we see that we have constructed a trigonometric polynomial 1 , 2 ( , ) which can approximate to on Ω very well.  is an odd function. By Theorem 1, we have ∈ ([−1, 1] 2 ) and ( + / )( , ) = 0 on ([−1, 1] 2 ) for 0 ≤ + ≤ . Again, doing a 2-periodic extension, we obtain a 2-periodic odd function and ∈ (R 2 ). By the well-known results [5,7,10], can be approximated by sine polynomials very well. Moreover, can be expanded into the Fourier sine series; that is, where the coefficients . Considering the approximation of by the partial sum, the Fejer sum, and the Vallee-Poussin sum [7,14] of the Fourier sine series of , we will obtain the approximation of the original function on Ω by sine polynomials.

Wavelet Analysis
1 be a bivariate smooth wavelet [2]. Then, under a mild condition, the families are a periodic wavelet basis. We expand into a periodic wavelet series [2] From this, we can realize the wavelet approximation of on Ω, for example, if = 2, its partial sum satisfies ‖ − 2 ( )‖ 2 ( ) = (2 −2 ). From this and ( , ) = ( , ) (( , ) ∈ Ω), we will obtain an estimate of wavelet approximation for a smooth function on the domain Ω.
(ii) Wavelet Approximation. Let be the smooth function with a compact support as in Theorem 3. Let be a univariate Daubechies wavelet and be the corresponding scaling function [2]. Denoting then { ( , )} 3 =1 is a smooth tensor product wavelet. We expand into the wavelet series where , , = 2 (2 ⋅ − ) and the wavelet coefficients Since is a smooth function, the wavelet coefficients , , decay fast.
For example, let * ∈ Z and * = ( By Lemma 8, we know that The Scientific World Journal If the Daubechies wavelet chosen by us is time smooth, then, by using the moment theorem and supp * , So 2 = 0. Similarly, since ( , ) is bivariate polynomials on rectangles 1 and 3 (see Lemma 11), we have 1 = 3 = 0. Furthermore, by (18), we get 1, * , * = 0. Therefore, the partial sum of the wavelet series (16) can approximate to very well and few wavelet coefficients can reconstruct . Since = on Ω, the partial sum of the wavelet series (16) can approximate to the original function on the domain Ω very well.

Proofs of the Main Theorems
We first give a partition of the complement \ Ω.

Ω ⊂
and Ω is a piecewise infinitely many time smooth curve, without loss of generality, we can divide the complement \Ω into some rectangles and some trapezoids with a curved side. For convenience of representation, we assume that we can choose four point ( ] , ] ) ∈ Ω (] = 1, 2, 3, 4) such that \Ω can be divided into the four rectangles and four trapezoids with a curved side where (24)  From this, we know that can be expressed into a disjoint union as follows: where each ] is a trapezoid with a curved side and each ] is a rectangle (see Figure 1). In Sections 3.2 and 3.3 we will extend to each ] and continue to extend to each ] such that the obtained extension satisfies the conditions of Theorem 1.
The first formula of (38) holds for . By (33), we have The second formula of (38) holds. By induction, (38) holds for all . From this, we get Lemma 5. Now we compute the mixed derivatives of ( ) 1 ( , ) on the curved side Γ 1 and bottom side Δ 1 of 1 .
By the Newton-Leibniz formula, we have From this and Lemma 5, it follows that, for any 1 ≤ ≤ 2 , we have Finding derivatives on the both sides of this formula, we get Similar to the argument from (46) to (50), we get Continuing this procedure, we deduce that (i) holds for 0 < + ≤ . Letting = 0 in Lemma 5, we have ( ) 1 ( , ( )) = ( , ( )); that is, (i) holds for = = 0. So we get (i).
From this, we get the following.
For ] = 2,3,4, by using a similar method, we define From this, we get (i) and (ii). The proof of (iii) is similar to the argument of Lemma 4(iii). Lemma 8 is proved.

Smooth Extension to Each Rectangle ] .
We have completed the smooth extension of to each trapezoid ] with a curved side. In this subsection we complete the smooth extension of the obtained function to each rectangle ] . First we consider the smooth extension of to 1 . We divide this procedure in two steps.

Representation of the Extension Satisfying Theorem 1
Let and Ω be stated as in Theorem 1 and let Ω be divided as in Section 3.1. The representation of satisfying conditions of Theorem 1 is as follows: It is also easy to check directly them. Again extend smoothly from [ 1 , 2 ] to [ 2 , 1], we construct two polynomials   Let ∈ ([ 1 , 2 ]) and [ 1 , 2 ] ⊂ (0, 1), and let be the smooth extension of from [ 1 , 2 ] to [0, 1] which is stated as in Theorem 12. Let be the 1-periodic extension satisfying ( + ) = ( ) (0 ≤ ≤ 1, ∈ Z). Then ∈ (R) and ( ) = ( ) ( ∈ [ 1 , 2 ]). We expand ( ) into the Fourier series which converges fast. From this, we get trigonometric approximation of ∈ ([ 1 , 2 ]). We also may do odd extension or even extension of from [0, 1] to [−1, 1], and then doing periodic extension, we get the odd periodic extension ∈ (R) or the even periodic extension ∈ (R). We expand or into the sine series and the cosine series, respectively. From this, we get the sine polynomial approximation and the cosine polynomial approximation of on [ 1 , 2 ]. For ∈ ( ) ( ∈ [0, 1]), we pad zero in the outside of [0, 1] and then the obtained function ∈ (R). We expand into a wavelet series which converges fast. By the moment theorem, a lot of wavelet coefficients are equal to zero. From this, we get wavelet approximation of ∈ ([ 1 , 2 ]).