The Scientific World Journal

Volume 2014 (2014), Article ID 102062, 16 pages

http://dx.doi.org/10.1155/2014/102062

Research Article

Approximation of Bivariate Functions via Smooth Extensions

Zhihua Zhang

College of Global Change and Earth System Science, Beijing Normal University, Beijing 100875, China

Received 10 December 2013; Accepted 26 December 2013; Published 10 February 2014

Academic Editors: N. I. Mahmudov and S. Wang

Copyright © 2014 Zhihua Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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.

2.2. Applications

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 [9]

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 [9]. 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 [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 , 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 [2]. 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 [2], 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).

Figure 1: Partition of the complement of the domain .

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.

3.2. Smooth Extension to Each Trapezoid with a Curved Side

By (23), the trapezoid with a curved side is represented as

We define two sequences of functions and as follows:

By (27), we deduce that for ,

On , we define a sequence of functions by induction.

Let

Then, by (27),

Let

Then, by (27)–(30), we obtain that, for ,

In general, let

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) , .

Proof. By the assumption , Lemma 4: , and Lemma 6(i):

where , we get (i). By Lemma 6(ii) and , we get (ii). Lemma 7 is proved.

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:

Proof. By Lemma 7, we have

Similar to the argument of Lemma 7, for , we have

From this, we get (i) and (ii).

The proof of (iii) is similar to the argument of Lemma 4(iii). Lemma 8 is proved.

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.

Step 1. In Lemma 8, we know that on . Now we construct the smooth extension of from to , where is stated in Section 4.2 and .

Let

and let

where .

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

By (72) and (73),

By induction, we get

From this and , we get (iii). By Lemma 6(ii) and Lemma 9(iii), we get that

From this and (72), by using an argument similar to the proof of (iii), we get (iv).

By Lemma 8(iii) and Lemma 9(i), we deduce that is a polynomial of degree with respect to . From this and (72), we get (v). Lemma 10 is proved.

By Lemmas 9 and 10, we obtain that for ,

Lemma 11. Let

Then one has

(i)(ii) for ;(iii), where each is constant.

Proof. By Lemma 7, we have . Since ,

by (81), we deduce that . Since ,

by (79), we deduce that . So we get (i).

By Lemma 8(ii),

Since , by (80), we deduce that

So we get (ii).

From Lemma 9(i), Lemma 10(v), and , we get (iii). Lemma 11 is proved.

For , by using a similar method, we define , where representations of and see Section 4.2.

3.4. The Proofs of the Theorems

Proof of Theorem 1. Let

By (25), has been defined on the unit square . The argument similar to Lemma 11(i)-(ii) shows that

and for ,

From this and , by (25), we have and . So we get (i) and (ii).

Similar to the argument of Lemma 11(iii), we get

where each is a constant. From this and Lemma 8(iii), we know that, on , can be expressed locally in the form

(iii) holds. We have completed the proof of Theorem 1.

The representation of satisfying the conditions of Theorem 1 is given in Section 4.

Proof of Theorem 2. Let be the smooth extension of from to which is stated as in Theorem 1. Define by

Then is a -periodic function of . By Theorem 1, we know that and

Let . Since is 1-periodic function, we have and for any ,

Noticing that , we have . By (92) and Theorem 1(i), we get

Theorem 2 is proved.

Proof of Theorem 3. Let be the smooth extension of from to which is stated as in Theorem 1. Define by

From Theorem 1(ii), we have

From this and (96), we get . By (96) and Theorem 1(i), we get

Theorem 3 is proved.

4. Representation of the Extension Satisfying Theorem 1

Let and be stated as in Theorem 1 and let be divided as in Section 3.1. The representation of satisfying conditions of Theorem 1 is as follows:

where

and the rectangles and the trapezoids with a curved side are stated in (22) and (23) and and .

Below we write out the representations of , , and .

4.1. The Construction of Each

(i) Denote

Define by induction as follows:

(ii) Denote

Define by induction as follows:

(iii) Denote

Define by induction as follows:

(iv) Denote

Define by induction as follows:

4.2. The Constructions of Each and

(i) Denote

Define by induction as follows:

Denote

Define by induction as follows:

(ii) Denote

Define by induction as follows:

Denote

Define by induction as follows:

(iii) Denote

Define by induction as follows:

Denote

Define by induction as follows:

(iv) Denote

Define by induction as follows:

Denote

Define by induction as follows:

5. Corollaries

By using the extension method given in Section 3, we discuss the two important special cases.

5.1. Smooth Extensions of Functions on a Kind of Domains

Let be a trapezoid with two curved sides:

where . Denote the rectangle . Then , where and are both trapezoids with a curved side:

Suppose that ( is a nonnegative integer). We will smoothly extend from to the trapezoids and with a curved side, respectively, as in Section 3.2, such that the extension function is smooth on the rectangle . Moreover, we will give a precise formula. It shows that the index of smoothness of depends on not only smoothness of but also smoothness of .

Denote and

We define on as follows. Let

and let be the maximal integer satisfying . For , we define

Then , where .

Denote and

We define on as follows. Let

For , define

Then , where is stated as above.

An argument similar to Lemmas 5 and 6 shows that, for and ,

A direct calculation shows that the number

is the maximal value of integers satisfying , where expresses the integral part. So .

By (133), we get that, for ,

Note that

and the assumption . Now we define a function on by

From this and (135), we have . This implies the following theorem.

Theorem 12. Let the domain and the rectangle be stated as above. If , then the function , defined in (137), is a smooth extension of from to and , where is stated in (134).

Especially, for and , we have , and so ; for and , we have , and so ; for and , we have , and so .

5.2. Smooth Extensions of Univariate Functions on Closed Intervals

Let and . In order to extend smoothly from to , we construct two polynomials

Define and for ,

Then is a polynomial of degree .

Similar to the proof of Lemma 5, we get

It is also easy to check directly them.

Again extend smoothly from to , we construct two polynomials

Define and for ,

Then is a polynomial of degree .

Similar to the proof of Lemma 5, we get

Therefore, we obtain the smooth extension from to by

where and are polynomials of degree defined as above, and and . From this, we get the following.

Theorem 13. Let and . Then there exists a function satisfying and .

Let and , and let be the smooth extension of from to which is stated as in Theorem 12. Let be the 1-periodic extension satisfying . Then and . We expand into the Fourier series which converges fast. From this, we get trigonometric approximation of . We also may do odd extension or even extension of from to , and then doing periodic extension, we get the odd periodic extension or the even periodic extension . 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 . For , we pad zero in the outside of and then the obtained function . 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 .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is partially supported by the National Key Science Program no. 2013CB956604; the Beijing Higher Education Young Elite Teacher Project; the Fundamental Research Funds for the Central Universities (Key Program) no. 105565GK; and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

References

I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Communications on Pure and Applied Mathematics, vol. 41, pp. 909–996, 1998.

I. Daubechies, Ten Lectures on Wavelets, vol. 6 of CBMS-Conference Lecture Notes, SIAM, Philadelphia, Pa, USA, 1992.

C. K. Chui, L. L. Schumaker, and J. Stockler, Approximation Theory X: Wavelets, Splines, and Applications, Vanderbilt University Press, 2002.

C. K. Chui, W. Schempp, and K. Zeller, Multivariate Approximation Theory, Birkhauser, 1989.

C. K. Chui, “Approximation induced by a fourier series,” Rocky Mountain Journal of Mathematics, vol. 4, no. 4, pp. 643–648, 1974.

R. Long, Multidimensional Wavelet Analysis, International Publishing, Beijing, China, 1995.

G. G. Lorentz, Approximation of Functions, Holt, Rinebart and Winston, 1966.

P. Petrushev, “Multivariate n-term rational and piecewise polynomial approximation,” Journal of Approximation Theory, vol. 121, no. 1, pp. 158–197, 2003.

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971.

A. F. Timan, Theory at Approximation of Functions of a Real Variable, Pergamon, 1963.

Z. Zhang, “Convergence and Gibbs phenomenon of periodic wavelet frame series,” Rocky Mountain Journal of Mathematics, vol. 39, no. 4, pp. 1373–1396, 2009.

Z. Zhang, “Measures, densities and diameters of frequency bands of scaling functions and wavelets,” Journal of Approximation Theory, vol. 148, no. 2, pp. 128–147, 2007.

D. Zhou and K. Jetter, “Approximation with polynomial kernels and SVM classifiers,” Advances in Computational Mathematics, vol. 25, no. 1–3, pp. 323–344, 2006.

A. Zygmund, Trigonometric Series, I, II, Cambridge University Press, 2nd edition, 1968.