Fourier Expansions with Polynomial Terms for Random Processes

Abstract

Based on calculus of random processes, we present a kind of Fourier expansions with simple polynomial terms via our decomposition method of random processes. Using our method, the expectations and variances of the corresponding coefficients decay fast and partial sum approximations attain the best approximation order. Moreover, since we remove boundary effect in our decomposition of random process, these coefficients can discover the instinct frequency information of this random process. Therefore, our method has an obvious advantage over traditional Fourier expansion. These results are also new for deterministic functions.

1. Introduction

Based on calculus of random processes [1], it is well known that one can expand a random process into Karhunen-Loeve series or Fourier series. For a Karhunen-Loeve series, its coefficients are random variables and its basis consists of deterministic functions which are eigenfunctions of the correlation function of the random process. So, when one analyzes different random processes, one needs to compute different Karhunen-Loeve basis. However, to compute eigenfunctions numerically is very difficult. Therefore, one often expands random processes into Fourier series. Due to the discontinuity on the boundaries, their Fourier coefficients decay slowly and cannot reflect the instinct frequency information of the random process. To solve this problem, we give a new method. First, we give the decompositions of the one-dimensional and two-dimensional random processes. Based on our decompositions, we present a kind of Fourier expansions with simple polynomial terms, in which Fourier coefficients decay fast. Although we discuss these expansions in the setting of random processes, for deterministic functions, our results are also new.

Our main idea is as follows.

For a one-dimensional random process on , we present a Fourier expansion with a polynomial term: as in (27). The first term is a random polynomial which is determined by the derivatives of at and . The second term is a Fourier sine series of . Theorem 2 shows that the expectations and variances of Fourier coefficients of decay fast and Theorem 4 shows that the partial sum approximation of expansion (27) attains the best approximation order.

For a two-dimensional random process on , we present a Fourier expansion with polynomial terms: as in (49). The first term is a two-dimensional random polynomial which is determined by the partial derivatives of at the vertices of the unit square . The second term contains four one-dimensional Fourier sine series which are determined by the partial derivatives of on the boundary of . The third term is the two-dimensional Fourier series of . Theorems 6 and 7 show that the expectations and variances of all the above Fourier coefficients in expansion (49) decay fast. The partial sum of expansion (49) is a combination of random polynomials and random sine polynomials. This is a good approximation tool. Theorem 9 shows that the partial sum approximation attains the best approximation order. In this combination, degrees of these random polynomials are only determined by smoothness indexes of the random process while the degrees of sine polynomials are determined by presumed approximation error and smoothness indexes.

These results explain that the Fourier expansion with polynomial terms is better than the Fourier expansion. Our results are also new for deterministic functions.

This paper is organized as follows. In Section 2 we recall some known results in the basic calculus of random processes. In Section 3 we give the decompositions of the one-dimensional and two-dimensional random processes. In Sections 4 and 5 we study Fourier expansions with polynomial terms for one-dimensional and two-dimensional random processes, respectively. In Section 6 we state our conclusions.

2. Calculus of Random Processes

Calculus of random processes is the basis of random differential equations and has a wide application in many fields, such as finance, economy, and geoscience [1–8].

If a real random variable is such that the expectation of its square is finite, then we say that it is a second order random variable. Let be a sequence of second order random variables and let be a second order random variable. If we say converges to in the mean square sense. Based on it, one can derive the concepts of the continuity, the derivatives, and the integrals as well as partial derivatives and double integrals of random process [1, 7, 8]. If the coefficients of a polynomial are random variables, it is said to be a random polynomial. The -degree derivative of a one-dimensional random polynomial of degree is a random variable which is independent of , and its -degree derivative vanishes. If a random process is continuous on , one says that it belongs to . By and denote the expectation and variance of a random process , respectively. If is a continuously differentiable random process on , then And the Newton-Leibnitz formula holds. For the product of a differentiable random process and a differentiable deterministic function on , the product differential method holds and the corresponding integration by parts also holds. However, it does not hold for the product of two differentiable random processes. Based on these concepts, theory on Karhunen-Loeve expansions or Fourier expansions of random processes is developed.

Throughout this paper, we denote the vertices of by and its boundary by . The integral part of a positive real is denoted by . Denote the differential operators by , where , are nonnegative integers. means that and means that is bounded.

3. Decompositions of Random Processes

We assume that is a random process on and its -order derivatives . If we expand it into Fourier sine series the expectations of its Fourier sine coefficients decay slowly. In fact, for , By the Riemann-Lebesgue lemma [9] and (4), so we see that if and only if . This means that the decaying of Fourier sine coefficients depends on the value of the random process at the endpoints. Inspired by this fact, we consider removing a simple random polynomial from the random process such that the expectations and variances of Fourier sine coefficients of the residual decay fast. For this purpose, let be a polynomial of degree and their derivatives satisfy Then can be represented as follows: where the coefficients for satisfy We construct a random polynomial as follows: We get the decomposition formula .

By (9) and (12), we have In Theorem 2, we will show that the expectations and variances of Fourier sine coefficients of decay fast.

Let be a two-dimensional random process on and . We decompose it as follows: where

Theorem 1. Let be a random process on and and let it be decomposed as , where , , and are stated in (15)–(18). Then, for , , one gets the following.
(i) is a two-dimensional random polynomial and where .
(ii) and where ().

Proof. From (9), it follows by (15) that Similarly, we discuss all vertices of the unit square and we get (i).
By and (i), we get that Similarly, we have .
By (9), we get where (), (). By (16), we have By , we have Similarly, we discuss each side of the unit square and we get (ii).

4. One-Dimensional Case

Let be an -order differentiable real random process on . By the decomposition formula, we have , where is stated in (12). Expanding into Fourier sine series, we obtain the Fourier expansion of with a polynomial term,

First, we estimate the expectations and variances of the Fourier sine coefficients .

Denote (i) The Case That Is an Odd Number. In (26), since is an -order differentiable random process and is a differentiable deterministic function, using the integration by parts times, we get by (13) that Since the expectation and the integral can be exchanged, by (29), By , we have . From this and the Riemann-Lebesgue lemma, we get Noticing that is an odd number and , we have . Again, since is a random polynomial of degree , we know that is a random variable which is independent of and is denoted by . So we deduce that From this and , we get by (29) that From this and (28), we have Again, by the Riemann-Lebesgue lemma, . By the Schwarz inequality in the probability theory and (28), we have So we get .

(ii) The Case That Is an Even Number. By , we have . Since is a random polynomial of degree , we have (). Again, use the formula of integration by parts times to get With the help of a similar argument, we also deduce that From a known formula, we have , so .

Summarizing the above results, we get the following.

Theorem 2. Let be a random process on and and let be expanded as in (27). Then Fourier sine coefficients satisfy where , are stated as in (28).

Remark 3. Under the condition of Theorem 2, if we directly expand into the traditional Fourier series or Fourier sine series, it is easy to prove that the expectations and variances of coefficients satisfy Next, we consider the partial sum approximation of expansion (27). We define its partial sum as where and the second term of the right-hand side is the partial sum of the Fourier sine series (26) of .
The partial sum is a combination of an -degree random polynomial and an -order random sine polynomial. This is a good approximation tool and can attain the best approximation order for -degree continuously differentiable random processes on .
In fact, by (27) and (41), . Denote . By the Parseval identity, By Theorem 2, we have , so So we get the following.

Theorem 4. Let be an -order continuously differentiable random process on and let be the partial sum of its Fourier expansion with polynomial terms which is stated in (41). Then the mean square error where is stated in (28).
Theorem 4 gives the best approximation order for -order continuously differentiable random processes on even if it can be smoothly extended to a 2-periodic random process with the same smoothness, its approximation order is just .

Remark 5. Under the condition of Theorem 4, it is easy to prove that the partial sum of the Fourier expansion or Fourier sine expansion of only satisfies .
Comparing Theorems 2 and 4 with Remarks 3 and 5, we see that the Fourier expansion with a polynomial term is better than the Fourier expansion.

5. Two-Dimensional Case

We will extend the one-dimensional results to the two-dimensional case.

5.1. Fourier Sine Expansion with Polynomial Terms

Let be a two-dimensional real random process and . By the decomposition formula of two-dimensional random processes, we have , where , , and are stated as in (15)–(18). For convenience, in this decomposition, we denote where . We expand the one-dimensional processes , , , and into the one-dimensional Fourier sine series, and expand the two-dimensional process into the two-dimensional Fourier sine series, where We finally expand into the Fourier sine series with polynomial terms as follows: where are the fundamental polynomials stated in Section 3.

Now we estimate the expectations and variances of these Fourier sine coefficients: , (), and . Denote Note that Thus . By Theorem 1(i), we obtain that, for , Similarly, .

First, we assume that the index is an odd number. By the integration by parts, Since , we have by (45) that Again, by (10) and (15), we know that is a random polynomial of degree with respect to . Therefore, is a random variable which is independent of , so we have From this and (53), we deduce that, for , So, by (50), and where . By the Schwarz inequality and (50), we have So ().

Now we assume that the index is an even number. Then Similarly, we also can get that, for , Again, since , we deduce that

Similarly, we discuss the coefficients , , and . Finally, we deduce the following.

Theorem 6. Let be a random process on and and let , be stated as in (45). Then, for and , where and are stated in (50).
By Theorem 1(ii), we know that satisfies and . If and are both odd numbers, using the integration by parts, we get where which is the Fourier cosine coefficients of on .
From , it follows that First, we consider the Fourier coefficients of . Noticing that is a polynomial of degree , we know by (15) that is a random variable which is independent of , . So we have By (16), we have By (18) and (10), we deduce that vanishes and where . Similarly, we have So we get . From this, (66), and (67), So, by (64), we get So By the Schwarz inequality and (50), we get so From this and , we have We easily check that whether and are odd or even, this formula and (73)–(75) always hold. Again, by the Riemann-Lebesgue lemma, we have the following.

Theorem 7. Let be a random process on and and let be expanded as in (49). Then the Fourier coefficients satisfy where and are stated as in (50).

Remark 8. Under the condition of Theorem 7, if we directly expand into the traditional Fourier series or Fourier sine series, it is easy to prove that the expectations and variances of coefficients only satisfy

5.2. A Good Approximation Tool of Random Processes

We assume that is a real random process on and . We expand it into Fourier sine series with polynomial terms (49). Replace “” on the right-hand side of (49) by to obtain partial sums of expansion (49), denoted by , where is a two-dimensional random polynomial and and is a sum of four one-dimensional random sine polynomials with polynomial factors and a two-dimensional random sine polynomial, and where is the th partial sum of the Fourier sine series of and

The partial sum is a combination of a random polynomial and random sine polynomials with polynomial factors . It is a good approximation tool for an -order differentiable random process and can attain the best approximation order. In this combination, the degree of the random polynomial is determined by smoothness index of random process, while the degrees of random sine polynomials are determined by the presumed approximation error.

In fact, from this and (49), we get Denote . By the inequality , we have Noticing that is a product of separation variables, we deduce that Since is a polynomial of degree , by (9), we have By (10), , so By the Parseval identity in the space , we have By Theorem 6 and the Parseval identity, we have From this, (84), and (85), we have Similarly, we get