CORRECTED FOURIER SERIES AND ITS APPLICATION TO FUNCTION APPROXIMATION

Any quasismooth function f(x) in a finite interval [0,x0], which has only a finite number of finite discontinuities and has only a finite number of extremes, can be approximated by a uniformly convergent Fourier series and a correction function. The correction function consists of algebraic polynomials and Heaviside step functions and is required by the aperiodicity at the endpoints (i.e., f(0)≠f(x0)) and the finite discontinuities in between. The uniformly convergent Fourier series and the correction function are collectively referred to as the corrected Fourier series. We prove that in order for the mth derivative of the Fourier series to be uniformly convergent, the order of the polynomial need not exceed (m


Introduction
The theory about the function approximation of finite functions in a finite interval by a Fourier series emerged as early as the nineteenth century [2].In particular, for any quasismooth function in [0,x 0 ], which is referred to as the single-valued finite function f (x) that has only a finite number of finite discontinuities as well as a finite number of maxima and minima in the interval [0,x 0 ], one could obtain the following Fourier series (partial sum): where the series coefficient A n is the Fourier projection of f (x) to the basic function e iαnx , that is, Notice that different definitions for smoothness of a function have been used with different meanings in different contexts, for example, in statistics [4].The concern here is whether the Fourier series (1.1) converges to f (x).The well-known Dirichlet theorem [2] states that for any quasismooth function, x ∈ 0,x 0 , f (0) + f x 0 2 x = 0 or x 0 . (1.3) If f (x) is aperiodic (i.e., f (0) = f (x 0 )) or there are discontinuities in (0, x 0 ), the Fourier series will not uniformly converge to f (x).The so-called Gibbs phenomenon appears near the endpoints and those discontinuities [3].On the other hand, if f (x) is a periodic quasismooth function without discontinuities, the Fourier series (1.1) is uniformly convergent to f (x) without the Gibbs phenomenon.The Gibbs phenomenon is artificial oscillations near the discontinuities and aperiodic endpoints.They are numerical noise without any physical meaning, and should be eliminated if possible.In [1], the Fourier series with Gibbs oscillations is reexpanded into a Gegenbauer series.By doing so, they effectively filter out the Gibbs oscillation and obtain a rapidly convergent series.In this study, we will show that by using a correction function we directly obtain a uniformly convergent Fourier series without Gibbs oscillation (Section 2).We will refer to the correction function and Fourier series as the corrected Fourier series.Since the corrected Fourier series is uniformly convergent, we will apply it to the function approximation, as illustrated in Section 3. Section 4 then provides the concluding remarks.

Corrected Fourier series
As stated in the introduction, a quasismooth function f (x) in [0,x 0 ] has only a finite number of finite discontinuities as well as a finite number of maxima and minima in the interval [0,x 0 ].Relevantly, a quasismooth continuous function is referred to as the quasismooth function without any discontinuity within the interval, but it can be either periodic or aperiodic.
Furthermore, two classes of functions are defined as follows.One is the class of mth quasismooth functions Q m ([0,x 0 ]), of which the mth derivative of each member is a quasismooth function in [0, x 0 ].Another is the class of mth quasismooth continuous functions S m ([0,x 0 ]), of which the mth derivative of each member is a quasismooth continuous function in [0,x 0 ].Here m ≥ 0. In the case of m = 0, Q 0 ([0,x 0 ]) has its member being a quasismooth function and S 0 ([0,x 0 ]) being a quasismooth continuous function.The extension of Q m ([0,x 0 ]) and S m ([0,x 0 ]) to the cases with multiple variables is straightforward; for two variables they are denoted as Q m ([0,x 0 ],[0, y 0 ]) and S m ([0,x 0 ],[0, y 0 ]), respectively.
In addition, an mth uniformly convergent Fourier series in an interval means that the Fourier series remains uniformly convergent until its mth derivative without Gibbs phenomenon.

Qing-Hua Zhang et al. 35
For clarity, the following notations will be used: (2.1) Heaviside step function Ᏼ(x − x j ) is defined as (2.2) ) can be partitioned into an mth quasismooth continuous function f m (x) ∈ S m ([0,x 0 ]) and mth integrals of a set of Heaviside step functions as follows: where x j is one of the discontinuities of the mth derivative of g m (x), denoted as In (2.3), the mth derivative of f m (x), denoted as f (m) m (x), is continuous at the discontinuities of g (m)  m (x).This can be easily shown: ) can be approximated uniformly by the sum of an mth uniformly convergent Fourier series and a polynomial no more than (m + 1)th order:

.5)
Proof.We start with m = 0. Any function f 0 (x) ∈ S 0 ([0,x 0 ]) is a quasismooth continuous function.It can be expressed by a periodic quasismooth continuous function h(x) and a linear function as follows: where a = [ f 0 (x 0 ) − f 0 (0)]/x 0 .It is easy to see the periodicity of h since h(0) = h(x 0 ) = f 0 (0).Because h(x) is a periodic, quasismooth continuous function, it can be approximated by a Fourier series that is uniformly convergent: where It follows that A n e iαnx + ax. (2.9) Notice that the right-hand side of the above equation has two parts: one is the Fourier series and other is a 1st-order polynomial.In other words, the function f 0 (x) is represented by a corrected Fourier series.Hence, the theorem is true for m = 0.In the next step, consider the 1st quasismooth continuous function f 1 (x) ∈ S 1 ([0,x 0 ]).Because its first derivative f (1)  1 (x) is a quasismooth continuous function in the interval [0,x 0 ], f (1)  1 (x) can be uniformly approximated by a corrected Fourier series: A n e iαnx + A 0 + ax.
(2.10) Its integration yields where (2.12) Since the termwise integration of a series improves its convergence, the implied uniformly convergence in the above equation is valid.Thus, the theorem is valid for m = 1.
According to the axiom of the mathematical induction, our proof will be complete if the theorem can be proved true for any f m+1 (x) ∈ S m+1 ([0,x 0 ]) after it is assumed true for any f m (x) ∈ S m ([0,x 0 ]).
By definition, the 1st derivative of f m+1 (x) is an mth quasismooth continuous function in S m ([0,x 0 ]), for which the theorem has been assumed true.Consequently, we have A n e iαnx + A 0 + m+1 l=1 a l x l l! .
(2.13) Its integration yields where (2.15) The right-hand side of (2.14) is the same as that of (2.5).This means that the theorem is true for any f m+1 (x) ∈ S m+1 ([0,x 0 ]) after it is assumed true for any f m (x) ∈ S m ([0,x 0 ]).
Corollary 2.3.Any mth quasismooth function g m (x) ∈ Q m ([0,x 0 ]) can be uniformly approximated by a corrected Fourier series consisting of three parts: an mth uniformly convergent Fourier series, a no-more-than (m + 1)th-order polynomial, and an mth integral of the Heaviside step functions at the discontinuities It is easily seen that the corollary holds right after Lemma 2.1 and Theorem 2.2.

Function approximation: examples
As shown in the preceding section, the corrected Fourier series is uniformly convergent.
In this section, we will make use of the corrected Fourier series to uniformly approximate quasismooth functions.Two 2nd quasismooth continuous functions in S 2 ([0,x 0 ]) and S 2 ([0,x 0 ],[0, y 0 ]), respectively, have been chosen to show how to determine coefficients of the corrected Fourier series.Extending to higher-order functions should be straightforward but tedious.

2nd quasismooth continuous function f (x)
∈ S 2 ([0,x 0 ]).According to the theorem in Section 2, the uniformly convergent corrected Fourier series of f (x) is in the form of and its 1st and 2nd derivatives are correspondingly As implied, both derivatives of the corrected Fourier series are uniformly convergent.
Based on the endpoints values of f (x) and its derivatives (endpoints effect), we obtain the following linear equations for a l (l = 1,2,3): 3) are zero.The corrected Fourier series is just the regular one, which is uniformly convergent in the interval [0, x 0 ].The coefficients a l (l = 1,2,3) are easily obtained by solving (3.3):

.4)
The coefficient A n in (3.1) is the Fourier project (1.2) of f (x) − (a 1 x + a 2 (x 2 /2!) + a 3 (x 3 /3!)) on the basic function e iαnx , that is, where If any of a l (l = 1,2,3) is nonzero or f (x) is aperiodic, then Gibbs oscillations are expected in n Ᏺ 1 f (x) n e iαnx .However, the same oscillations exist in 3  l=1 n a l I ln e iαnx .
Qing-Hua Zhang et al. 39 They cancel each other exactly in n A n e iαnx = n Ᏺ 1 f (x) n e iαnx − 3 l=1 n a l I ln e iαnx .As a result, n A n e iαnx is free of Gibbs oscillations and is uniformly convergent.
The next task is to determine a lm , b ln , and A nm .In addition to the Fourier projection Ᏺ 1 • n (1.2), two additional Fourier projections are defined: (3.10)With respect to x, the endpoints effects of f (x, y) and its derivatives yield Notice that the first and third terms in (3.8) are exactly zero due to the periodicity of e iαnx .After the Fourier projection to e −iβm y (i.e., Ᏺ 2 • m ), the following equations are resulted and are solved for a lm (l = 1,2,3) for each m: where Similarly, the endpoints effect of f (x, y) and its partial derivative with respect to y result in the following linear equations for b ln for each n: where Qing-Hua Zhang et al. 41 After d ll0 , a lm , and b ln are obtained, the rest of the coefficients A nm are readily obtained by the following Fourier projection: Just as in the first example, any possible Gibbs oscillations associated with the partial sum of the first term in the above equation will be canceled exactly by those in the square bracket.After all, each coefficient in (3.8) has been determined.This completes the function approximation of f (x, y) by a corrected Fourier series.

Concluding remarks
Any quasismooth function can be uniformly approximated by a corrected Fourier series, which consists of a uniformly convergent Fourier series and a correction function.The corrected Fourier series is free of the Gibbs phenomenon, although the quasismooth function can be aperiodic and have discontinuities in general.
The correction function consists of algebraic polynomials and Heaviside step functions.The orders of the polynomials are no more than (m + 1), demanding that the mth derivative of the corrected Fourier series be uniformly convergent.The corrected Fourier series will not be overconstrained, if the function to be approximated has defined its derivation only until mth order.The solution of an mth-order ordinary or partial differential equation is one such function whose (m + 1) derivative is not necessarily defined.Applications of the corrected Fourier series to linear ordinary differential equations with varying coefficients and to linear partial differential equations on irregular region will be the subject of our future studies.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning the boundary values of f (x, y) and its 1st and 2nd partial derivatives only.Notice that the first three terms on the right-hand side of(3.8)are identically zero due to the periodicity of either e iαnx or e iβm y .If we arrange the nine unknowns into a vector ordered as (d 11 ,d 12 ,d 13 ,d 21 ,d 22 ,d 23 ,d 31 ,d 32 ,d 33

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation