Approximation of Functions on a Square by Interpolation Polynomials at Vertices and Few Fourier Coefficients

Abstract

For a bivariate function on a square, in general, its Fourier coefficients decay slowly, so one cannot reconstruct it by few Fourier coefficients. In this paper we will develop a new approximation scheme to overcome the weakness of Fourier approximation. In detail, we will use Lagrange interpolation and linear interpolation on the boundary of the square to derive a new approximation scheme such that we can use the values of the target function at vertices of the square and few Fourier coefficients to reconstruct the target function with very small error.

1. Introduction

It is well-known that smooth periodic functions can be approximated well by Fourier series [1, 2]. But, if we expand a bivariate function on the square robustly into Fourier series, in general, its Fourier coefficients decay slowly. Therefore, one cannot reconstruct it by few Fourier coefficients [3–5]. This is also viewed as the weakness of Fourier approximation compared with wavelet approximation [6, 7]. In this paper we will develop a new approximation scheme to overcome the weakness of Fourier approximation. Our main idea is as follows: For a smooth function on the square , based on the Lagrange interpolation and the linear interpolation [8–11] of the target function at the vertices of , we construct a bivariate function such that on the boundary of . Secondly, we take Fourier expansion of the residual . Finally, based on this decomposition, we can derive a new scheme to approximate by using the values of at the vertices of and few Fourier coefficients.

Throughout this paper we denote the boundary of the unit square by and denote the vertices of by ; that is, Denote and . We say if .

Expand into a Fourier series: where is the Fourier coefficient of : The Fourier series hyperbolic cross truncations are defined as [12]

2. New Approximation Scheme of Bivariate Functions

Let be defined on the square . We use its values at vertices of the square and its few Fourier coefficients to reconstruct on .

Step 1. Take the linear interpolation of on the vertical intercept of the square , for a fixed : Let . Expand into Fourier series: where is the Fourier coefficient of . We reconstruct by Let . Expand into Fourier series: We reconstruct by So we reconstruct by that is, the values of at vertices and Fourier coefficients .

Step 2. Take the linear interpolation of on the horizontal intercept of the square , for a fixed , Let . Expand into Fourier series: where is the Fourier coefficient of . We reconstruct by Let . Expand into Fourier series: We reconstruct by So we reconstruct by that is, the values of at vertices and Fourier coefficients .

Step 3. Take the bivariate interpolation polynomial of with nodes : Define an algebraic sum as which can be reconstructed by .

Step 4. Let Expand into Fourier series: Take the hyperbolic cross truncation of the Fourier series: which can reconstruct by , that is, by few Fourier coefficients. Precisely say that the number of Fourier coefficients used in is equal to ; that is,

Step 5. We reconstruct the target function by Now we can reconstruct a target function by the values of at : and Fourier coefficients , , and .

3. Asymptotic Formula of the Errors for the Reconstruction of

In order to give the error estimate of our approximation scheme, we need to first give the asymptotic formula of the Fourier coefficients.

Assume that . In Section 2, we know that . We first reconstruct . To reconstruct it, we need to use Fourier coefficients , and which are stated in Section 2. From we deduce that . So Since , we have , and so This gives an asymptotic formula as follows: Differentiating (24), we get . So

Similarly, we discuss the Fourier coefficients , and . We get the following.

Theorem 1. Let and , , , and be stated in Section 2. Then, for their Fourier coefficients, one has the following asymptotic formulas:

Denote the partial sums of the Fourier series of by ; that is, By the Parseval identity, the mean square error of satisfies From (37), it follows that Noticing that , we get Therefore, we can reconstruct by Fourier coefficients and we obtain the asymptotic formula of the error. Again, by Theorem 1, we get From this, we see that can be reconstructed by and Fourier coefficients , the order of the error is , and the coefficient is .

Similarly, for the boundary functions , and , the corresponding asymptotic formulas are the following:

By definitions of and in Section 2, we get So, from the above four asymptotic formulas, we know that where

From this, we know that the function , defined in (18), can be reconstructed by and and Fourier coefficients and , and the order of error is .

From the reconstruction of , we know that it is a simple combination of the boundary functions of and factors . It possesses the following important properties.

Theorem 2. Let be defined on and let be stated in Section 2. ThenThat is, and have same boundary functions: .

Proof. By (5), (11), and (17),So . Similarly, the rest equalities can be proved by the same way.

4. Asymptotic Formula of Approximation Error for

In this section, we will estimate the approximation errors of our proposed approximation scheme. In order to show that in (19) can be reconstructed by few Fourier coefficients, we give the following asymptotic formula of Fourier coefficients of .

Theorem 3. Let be defined on and let be stated as in Theorem 2, and letThen Fourier coefficients of satisfy the following asymptotic formula:

Proof. From , it follows thatFrom , we have , and so Since is a differentiable function of , the integral value in this formula is equal to ; that is,From and (5), (11), and (17), it can be checked that and soBy , we deduce that . From this, we get (42).

Theorem 3 shows that the decay rate of Fourier coefficients of is equivalent to . The principal part of only depends on the values of at vertices .

Now we discuss the special case: Fourier coefficients and .

Since and ,So

Now we compute .

By (5), (11), and (17) and Theorem 2, we haveand so From this and (49), it follows that , where

Similarly, we have (), where

By (52), (53), and Theorem 3, it follows thatwhere and are stated in (52) and (53).

To reconstruct by few Fourier coefficients, we consider the hyperbolic cross truncation of the Fourier series of :and soBy the Parseval identity, the mean square error satisfies whereNoticing that and , we deduce that From this and (57), we obtain finally the following theorem.

Theorem 4. Suppose that and be stated in (19) and be the hyperbolic cross truncation of Fourier series of . Then the mean square error of reconstruction of by satisfies the following asymptotic formula:where

5. Approximation Error of

In this section, we will compare our improved Fourier approximation scheme with traditional ones. By (18), (19), and (23),By (57) and (60),where , and are stated as above and is stated in (38).

To construct , data used by us are four values of at vertices on and univariate Fourier coefficients as well as bivariate Fourier coefficients , where Therefore, in the reconstruction scheme, the total number of coefficients used by us is less than . From this and (63), we get where is the number of coefficients used to construct .

One the other hand, if we directly expand into Fourier series, and use the partial sums of Fourier series as a approximation toolto reconstruct , then, from [3–5], we deduce that There are Fourier coefficients in the partial sum ; that is,Comparing this formula with (65), we see that the approximation scheme proposed in this paper has obvious advantage over traditional method.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research is partially supported by National Key Science Program no. 2015CB953602; Fundamental Research Funds for the Central Universities (Key Program) no. 105565GK; Beijing Young Talent Fund and Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.