Research Article | Open Access
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.
In the recent several decades, various approximation tools have been widely developed [1–14]. 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 higher-dimensional 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 and the interior of by and always assume that is a simply connected domain. We say that if the derivatives are continuous on for . We say that if all derivatives are continuous on for . We say that a function is a -periodic function if , where is an integer. We appoint that and the notation is the integral part of the real number .
2. Main Theorems and Applications
In this section, we state the main results of smooth extensions and their applications in Fourier analysis and wavelet analysis.
2.1. Main Theorems
Our main theorems are stated as follows.
Theorem 1. Let , where and the boundary is a piecewise infinitely many time smooth curve. Then for any there is a function such that (i);(ii) on the boundary for ;(iii)on the complement can be expressed locally in the forms where is a positive integer and each coefficient is constant.
Theorem 2. Let , where is stated as in Theorem 1. Then, for any , there exists a 1-periodic function such that .
Theorem 3. Let , where is stated as in Theorem 1. Then, for any , there exists a function with compact support such that .
In Sections 3 and 4, we give constructive proofs of Theorems 1–3. In these three theorems, we assume that . If ( is a nonnegative integer), by using the similar method of arguments of Theorems 1–3, we also can obtain the corresponding results.
Here we show some applications of these theorems.
2.2.1. Approximation by Polynomials
Let be the smooth extension of from to which is stated as in Theorem 1. Then and on . By , denote the set of all bivariate polynomials in the form . Then where is the norm of the space . The right-hand side of formula (2) is the best approximation of the extension in . By (2), we know that the approximation problem of by polynomials on a domain is reduced to the well-known approximation problem of its smooth extension by polynomials on the square [4, 10].
2.2.2. Fourier Analysis
(i) Approximation by Trigonometric Polynomials. Let be the smooth periodic extension of as in Theorem 2. Then and on . 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 , 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 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  where . By Theorem 2, we obtain that, for , 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 which can approximate to on very well.
(iii) Odd (Even) Periodic Extension. Let be the smooth extension of from to which is stated in Theorem 1. Define on by Then is an odd function. By Theorem 1, we have and on for . Again, doing a 2-periodic extension, we obtain a 2-periodic odd function and . 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 by sine polynomials.
Define on as follows: Then is an even function on . By Theorem 1, and on for . Again, doing a 2-periodic extension, we obtain a 2-periodic even function and . By the well-known result [5, 10], can be approximated by cosine polynomials very well. Moreover, can be expanded into the Fourier cosine series. Considering the partial sum, the Fejer sum, and the Vallee-Poussin sum [5, 7, 14] of the Fourier cosine series of , we will obtain the approximation of the original function on by cosine polynomials.
2.2.3. Wavelet Analysis
(i) Periodic Wavelet Series. Let be stated in Theorem 2. Let be a bivariate smooth wavelet . Then, under a mild condition, the families are a periodic wavelet basis. We expand into a periodic wavelet series  From this, we can realize the wavelet approximation of on , for example, if , its partial sum satisfies . From this and , we will obtain an estimate of wavelet approximation for a smooth function on the domain .
(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 . Denoting then is a smooth tensor product wavelet. We expand into the wavelet series where and the wavelet coefficients Since is a smooth function, the wavelet coefficients decay fast.
On the other hand, since , , a lot of wavelet coefficients vanish. In fact, when and satisfy , we have . Besides, by condition (iii) in Theorem 1, we know that is univariate or bivariate polynomials on . By the moment theorem , we know that more wavelet coefficients vanish.
For example, let and satisfy , where . Then we have By Lemma 8, we know that where and . So If the Daubechies wavelet chosen by us is time smooth, then, by using the moment theorem and , we have So . Similarly, since is bivariate polynomials on rectangles and (see Lemma 11), we have . Furthermore, by (18), we get .
Therefore, the partial sum of the wavelet series (16) can approximate to very well and few wavelet coefficients can reconstruct . Since on , the partial sum of the wavelet series (16) can approximate to the original function on the domain very well.
3. Proofs of the Main Theorems
We first give a partition of the complement .
3.1. Partition of the Complement of the Domain in
Since and is a piecewise infinitely many time smooth curve, without loss of generality, we can divide the complement into some rectangles and some trapezoids with a curved side. For convenience of representation, we assume that we can choose four point such that can be divided into the four rectangles and four trapezoids with a curved side where , and and 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).
3.2. Smooth Extension to Each Trapezoid with a Curved Side
On , we define a sequence of functions by induction.
Let Then, by (27),
Lemma 4. For any , one has and
Proof. Since and , and , by the above construction, we know that for any .
For , since (34) holds. We assume that (34) holds for ; that is, This implies that Again, notice that and are polynomials of whose degrees are both . From this and (33), it follows that (34) holds for . By induction, (34) holds for all . Lemma 4 is proved.
Below we compute derivatives on the curved side and the bottom side of .
Lemma 5. Let be stated as above. For any , one has
Proof. By (30), We have known that, for , (38) holds.
Now we assume that (38) holds for .
For , by (33), we have For , by the assumption of induction, we have By (28), we have . So we get For , note that and . By (39), we get The first formula of (38) holds for .
By (33), we have For , by the assumption of induction and (28), we have and . So For , since , by (43), 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 on the curved side and bottom side of .
Lemma 6. Let and be the curved side and the bottom side of , respectively. Then, for , (i)(ii),where .
Proof. Let . Then we have
By the Newton-Leibniz formula, we have
Similarly, replacing by in this formula, we have
From this and Lemma 5, it follows that, for any , we have
Finding derivatives on the both sides of this formula, we get
Now we start from the equality Similar to the argument from (46) to (50), we get Continuing this procedure, we deduce that (i) holds for . Letting in Lemma 5, we have ; that is, (i) holds for . So we get (i).
By Lemma 5, . From this and , we have so (ii) holds. Lemma 6 is proved.
From this, we get the following.
Lemma 7. For any positive integer , denote . Let Then (i) and ; (ii) , .
For , by using a similar method, we define on the each trapezoid with a curve side. The representations of are stated in Section 4.1.
Lemma 8. For any , let where . Then, for , one has the following: (i);(ii) for ;(iii) can be expressed in the form:
3.3. 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 . We divide this procedure in two steps.
Lemma 9. Let be four sides of the rectangle Then one has the following(i), where is a constant;(ii);(iii);(iv),where .
Proof. By Lemma 8(iii), we have
So is a polynomial of degree with respect to . Since and are both polynomials of degree , (i) follows from (61).
Similar to the argument of Lemma 6, we get (ii) and (iv).
Since , by Lemma 7, we have By the definition of and (64), we have We assume that By (61), we get for , we have and . Again, by the assumption of induction, we get By (64), we have . From this and (67), we get Taking , we have Since , we get (iii). Lemma 9 is proved.
Step 2. In Lemma 8, we know that on . We consider the difference . Obviously, it is infinitely many time differentiable on since is a polynomial. Now we construct its smooth extension from to the rectangle as follows. Let and let From this, we obtain the following.
Lemma 10. possesses the following properties: (i) on ;(ii) on ;(iii) on ;(iv) on , where and are stated in (62);(v), where is a constant.
Proof. The arguments similar to Lemma 9(ii) and (iv) give the conclusions (i) and (ii) of this theorem. Now we prove (iii) and (iv).
By Lemma 6(i) and Lemma 9(ii), as well as , we get that, for , So we have
Now we assume that