Hyperbolic Cross Truncations for Stochastic Fourier Cosine Series

Abstract

Based on our decomposition of stochastic processes and our asymptotic representations of Fourier cosine coefficients, we deduce an asymptotic formula of approximation errors of hyperbolic cross truncations for bivariate stochastic Fourier cosine series. Moreover we propose a kind of Fourier cosine expansions with polynomials factors such that the corresponding Fourier cosine coefficients decay very fast. Although our research is in the setting of stochastic processes, our results are also new for deterministic functions.

1. Introduction

For approximations of multivariate functions by algebraic/trigonometric polynomials on full grids, the approximation rate deteriorates rapidly as the dimension increases [1, 2]; this is just so-called “dimension curse” problem. In order to solve it, hyperbolic cross approximations have received much attention in recent years [1, 3–5]. For multivariate periodic functions, Griebel and Hamaekers [4] discussed hyperbolic cross trigonometric approximation and gave the corresponding error estimate. Instead of trigonometric polynomial space on full grids, they used the hyperbolic cross space: as approximation space. In 2010, Shen and Wang [1] studied the hyperbolic cross Jacobi polynomial approximation for functions on the unit cube and gave various formulas on error estimates. Moreover, they also considered the hyperbolic cross Hermite/Laguerre polynomial approximation. In 2009, Boyd [6] deeply researched large-degree asymptotics for Fourier, Chebyshev, and Hermite coefficients of analytic functions. For high-dimensional polynomial interpolation approximation, one often uses tensor product method to generalize one-dimensional interpolation polynomial approximation with Chebyshev knots. In 2000, Barthelmann et al. [3] showed that when the dimension is larger than , it is suggested to use sparse grids instead of full grids. Moreover, Barthelmann et al. obtained the corresponding error estimates.

Up to now, for various hyperbolic cross approximation, the asymptotic formulas of approximation errors are not available. In this paper, we will deeply study the hyperbolic cross approximation in Fourier cosine analyses and give a precise asymptotic formula of approximation error of hyperbolic cross truncation. Although our research is in the setting of stochastic processes, our results are also new for deterministic functions.

In order to obtain asymptotic formulas of approximation errors, we first decompose stochastic process on into a sum of three terms , where is a stochastic polynomial determined by partial derivatives of at vertexes of , is determined by partial derivative values of on the boundary of and is a sum of four univariate stochastic processes with simple polynomial factors, and is a bivariate stochastic process whose partial derivatives vanish on the boundary of .

Secondly, based on the above decomposition, we show that Fourier cosine coefficients of a bivariate stochastic process on can be approximated asymptotically by a combination of partial derivatives of at vertexes of and univariate cosine coefficients on the boundary of .

Thirdly, for hyperbolic cross approximation, we give the following precise result: if is a stochastic process on with smoothness index , then its hyperbolic cross truncations (see (16)) of stochastic Fourier cosine series of satisfy the following asymptotic formula: where and is the mathematical expectation.

Finally, based on our decomposition of stochastic processes, we propose Fourier cosine expansions with polynomial factors (see (111)) whose hyperbolic cross truncations are a combination of stochastic algebraic polynomials and stochastic cosine polynomials. When the smoothness index of the stochastic process, for partial sum approximation, hyperbolic cross approximation, and hyperbolic cross approximation with polynomial factors, the square of their approximation errors are respectively, where is the number of Fourier cosine coefficients used in each approximation method. From this, we see that hyperbolic cross approximation with polynomial factors can reconstruct the stochastic process on by using the least Fourier cosine coefficients.

This paper is organized as follows. In Section 2 we recall stochastic calculus and stochastic Fourier cosine series. In Section 3 we give decompositions of stochastic processes. In Section 4 we discuss univariate stochastic Fourier cosine analyses. In Section 5 we give an asymptotic formula of Fourier cosine coefficients for bivariate stochastic processes. In Section 6 we discuss partial sum approximations. In Section 7 we discuss hyperbolic cross approximations. In Section 8, we present the Fourier cosine series with polynomial factors and study its hyperbolic cross approximations.

2. Fourier Cosine Series of Stochastic Processes

We recall some concepts in calculus of stochastic processes and stochastic Fourier cosine series.

2.1. Calculus of Stochastic Processes

For a stochastic variable , we denote its expectation, second-order moment, and variance by , , and , respectively. If is a stochastic variable for each , then we say is a stochastic process on . In this paper, we always assume that a stochastic process is real-valued and satisfies for each . This ensures that its expectation, variance, and second-order moment always exist. Calculus of stochastic processes is a generalization of classical calculus. Let be a sequence of stochastic variables and let be a stochastic variable. If , we say that is the limit of . Starting from the concept of the limit, one defines continuity, derivatives, partial derivatives, integrals, and double integrals of stochastic processes [7]. Moreover, Newton-Leibnitz formula in calculus of stochastic processes is as follows. If is a differentiable stochastic process on and the derivative is continuous on , then Let be a continuously differentiable stochastic process on and let be a continuously differentiable deterministic function on . Then [7] Burkardt et al. studied stochastic partial differential equations in [8]. Xiu reviewed the current state-of-the-art of numerical method for stochastic computations in [9].

2.2. Fourier Cosine Series

If is a univariate stochastic process on and , then it can be expanded into the Fourier cosine series in mean square sense; that is, where The corresponding Parseval identity is

If is a bivariate stochastic process on and , then it can be expanded into the Fourier cosine series in mean square sense, where Fourier sine coefficients are stochastic variables and The corresponding Parseval identity holds:

2.3. Partial Sums and Hyperbolic Cross Truncations

Let be a stochastic process on . Partial sums of its Fourier cosine series are The number of Fourier cosine coefficients in the partial sum is .

Hyperbolic cross truncations of its Fourier cosine series are where means the integral part. The number of Fourier cosine coefficients in the hyperbolic cross truncation is of order . The hyperbolic cross approximations have been widely used in multivariate function approximation [1, 3–5].

2.4. Some Notations

For convenience, we denote vertexes of the unit square by , the boundary of by .

For a bivariate stochastic process , denote its mixed derivative by . The notation represents the set of continuous stochastic processes on . If , then we say has the smoothness index on .

Let and be two sequences. If for any , then we say ; if for any , then we say ; here are constants independent of . If , , and as , then we say . If and , and as , then we say .

3. Decomposition of Stochastic Processes

In order to study stochastic Fourier cosine series, we give a decomposition of stochastic processes. Although this decomposition is given in the setting of stochastic processes, it is also new for deterministic functions.

Let be a bivariate stochastic process on and for some . First, based on a fundamental polynomial , we construct a stochastic polynomial as follows: This stochastic polynomial is determined by partial derivatives of at vertexes .

Denote The following is clear.

Proposition 1. Let be a bivariate stochastic process on with smoothness index and be stated in (18). Then

Now we define and we derive the following.

Proposition 2. Let be a bivariate stochastic process on with smoothness index and be stated in (21). Then

Proof. Consider the bottom side , of the square . Using Proposition 1, it follows from (20) that . Therefore, by , we get . Similarly, vanishes on other sides of the square .

From (18) and (21), we get a decomposition of stochastic processes on as follows.

Let be a stochastic process on and for some . Then the decomposition holds, where , , and are stated in (17), (20), and (21), respectively.

4. Univariate Stochastic Fourier Cosine Series

Suppose that is a stochastic process on and the derivative for some . Its Fourier cosine coefficients are as follows: where Since the expectation and the integral can be exchanged and , using the Riemann-Lebesgue lemma [10–12], we get From (25), Again, by the Riemann-Lebesgue lemma, we get Since , we have . By (24) and (28), we get By the Schwarz inequality, Therefore, we have By the Parseval identity, the partial sums of its Fourier cosine series satisfy From this, we deduce that if and only if .

5. Asymptotic Representations of Bivariate Fourier Cosine Coefficients

Suppose that is a bivariate stochastic process on and for some . We expand into the Fourier cosine series and Fourier coefficients where is stated in (13).

Based on the decomposition (23) of bivariate stochastic processes, we have In order to obtain the asymptotic representation of Fourier cosine coefficients , we will precisely compute the first two terms and estimate the expectation and variance of the last term on the right-hand side of (35) as follows.

(i) For the first term , by the representation (17) of stochastic polynomial , we get that, for and , where is an algebraic sum of values of at and signs for addition and subtraction are determined by odevity of and . Since we get .

(ii) For the second term , by (20), we know that is the sum of products of separated variables.

Denote The Fourier cosine coefficients of are equal to where each is the Fourier cosine coefficient of the univariate stochastic process and each and each are both Fourier cosine coefficients of univariate deterministic functions and . A direct computation shows that, for and ,

(iii) For the last term , using the integration by parts, we deduce that By Proposition 2, , the interior integral is equal to So By , we have . This implies that and so Taking expectations on these two equations, we get It follows from that is a continuous function on and is a continuous on . By the Riemann-Lebesgue lemma, we have since .

From (i), (ii), and (iii), we get and the error satisfies (47), where is stated in (37) and each is stated in (38).

Now we further estimate these four univariate Fourier cosine coefficients .

Lemma 3. Let each be stated as in (38). Then Fourier cosine coefficients satisfy

Proof. By similarity, we only prove the case , since . By Proposition 1 and , we have . From this, we deduce that Fourier cosine coefficients satisfy

We will deduce these asymptotic representations of Fourier cosine coefficients.

Theorem 4. Let be a stochastic process on and for some , and let and be stated in (37) and (38), respectively. Then (i) for , , where  and satisfies (ii) for , , where  and satisfies (iii) for and , and satisfies

Proof. When , denote By (48), . Using Lemma 3 and (47), we deduce that and So we get (i). Similarly, we can get (ii).

From this, we can give asymptotic representations of expectation, second-order moment, and variance of Fourier cosine coefficients.

Corollary 5. Under conditions of Theorem 4, we have (i) for and ,  where is stated in (51) and “” is uniform for ;(ii) for and ,  where is stated in (53) and “” is uniform for .

Proof. By similarity, we only prove (58).
From Theorem 4 (i), we deduce that, for , So we get (58).

From Corollary 5 (iii), we know that These results cannot be improved as smoothness index increases.

6. Approximation of Partial Sums

Let be a stochastic process on and for some . We expand into the Fourier cosine series:

We give further the asymptotic representation of approximation error of partial sums. The partial sums of Fourier cosine series (62) are defined as Using the Parseval identity, we get (i)We compute . The first term on the right-hand side of the formula (64) can be decomposed into three sums  where

First, we consider the interior sum of with , By Corollary 5 (i), we deduce that where “” is uniform for and Again, by and (67), we obtain that where “” is uniform for . By the convergence of the series and (65), we have

By (69), we deduce that there exists a constant such that Again, by Lemma 3, we have and so the series converges, and denote its sum by . So

By (72), we get . Notice that and (see (61)). We have . This implies that Here and are stated in (69).

Similarly, Here “” is uniform for and From , we have . Finally, by (65), we get where , are stated in (77) and (78). (ii)We compute and in (64). By the decomposition formula (23), By (17) and , we have . By (39), we have From this, we get

Similar to (47), we have Again, by (81), we have This implies by (64) that

Similarly, we get From this and (80), we get by (64) the following.

Theorem 6. Let be a stochastic process on and for some . Then the partial sums of its Fourier cosine series satisfy where and are stated in (77), (78), (86), and (87).

7. Approximation of Hyperbolic Cross Truncations

Suppose that is a stochastic process on and for some . We consider hyperbolic cross truncations of its Fourier cosine series: By the Parseval identity and (61), (86), and (87), we have where . We rewrite in the form

We first consider . By Corollary 5 (i), we have where “” is uniform for and Therefore, Here where .

By using the Schwarz inequality in Probability theory, By Lemma 3, . Therefore, . From this, we have Substituting (95) and (97) into (94), we get that, for ,

Similarly, we have Denote . We get This implies that So we have

Similarly, we have where   . From this and (91), we deduce by (90) that where . Finally, we get by (90) that So we get the following.

Theorem 7. Suppose that is a stochastic process on and for some . Let be the hyperbolic cross truncations of its Fourier cosine series, which is stated in (89). Then the following asymptotic formula holds: where .

From Theorem 7, we know that . This result cannot be improved as the smoothness index increases.

By (89), the number of Fourier cosine coefficients in hyperbolic cross truncation is and . From this, we get

Since the number of Fourier cosine coefficients in the partial sum is , by Theorem 6, we have

8. Approximation of Hyperbolic Cross Truncations with Polynomial Factors

In order to use the least Fourier cosine coefficients to reconstruct stochastic processes, we introduce the Fourier cosine expansion with polynomial factors. Suppose that is a stochastic process on and for some . By using notation (38), the decomposition (23) can be rewritten in the form In this decomposition, we expand each into univariate Fourier cosine series and expand into bivariate Fourier cosine series. Finally, we obtain the Fourier cosine expansion of with polynomial factors: where . Notice that and , where . Using Proposition 1, we have and , and so Therefore, we have and Similarly, for ,, and , we get From this, we see that these univariate Fourier coefficients decay fast.

Now we show that bivariate Fourier cosine coefficients decay fast. By and Proposition 2, using the integration by parts, we get So we get

Take the hyperbolic cross truncations of the expansion (111), where each is the first terms partial sums of Fourier cosine series of and is the hyperbolic cross truncation of Fourier cosine series of bivariate stochastic process of which is stated in (21). The hyperbolic cross truncation is a combination of stochastic polynomials and cosine polynomials. By (111) and (117), Again, by the formula , we get By using Parseval identity and (114), we get By Proposition 2, we have . Based on (114) and (116), the argument similar to Theorem 7 shows that From this and (119), we deduce that . Noticing that the number of Fourier cosine coefficients in , we get the following.

Theorem 8. Suppose that is a stochastic process on and for some . Then where is the number of Fourier cosine coefficients in .

Comparing Theorem 8 with (108) and (109), we obtain the following.

Theorem 9. Let be a stochastic process on and for some . Then where and are partial sums and hyperbolic cross truncations of Fourier cosine series of   , respectively, is the hyperbolic cross truncations of Fourier expansion of with polynomial factors, and is the number of Fourier cosine coefficients in each sum.

From this, we see that if we reconstruct by , we need the least Fourier cosine coefficients.

Conflict of Interests

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

Acknowledgments

This research is supported by National Key Science Program for Global Change Research no. 2013CB956604 and no. 2010CB950504; the Beijing Higher Education Young Elite Teacher Project; Fundamental Research Funds for the Central Universities (Key Program) no. 105565GK; and Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.