_{1}

The key objective of this paper is to improve the approximation of a sufficiently smooth nonperiodic function defined on a compact interval by proposing alternative forms of Fourier series expansions. Unlike in classical Fourier series, the expansion coefficients herein are explicitly dependent not only on the function itself, but also on its derivatives at the ends of the interval. Each of these series expansions can be made to converge faster at a desired polynomial rate. These results have useful implications to Fourier or harmonic analysis, solutions to differential equations and boundary value problems, data compression, and so on.

There is perhaps no better way starting the discussion than quoting directly from Iserles and Nørsett [

Despite what has been said, the Fourier series will lose much of its luster when used to expand a sufficiently smooth nonperiodic function defined on a compact interval. It is well known that a continuous function can always be expanded into a Fourier series inside the interval (the word “inside” is highlighted to emphasize the fact that the two end points shall not be automatically included). This is actually the primary reason for the inefficiency of the Fourier series in approximating a nonperiodic function, and, understandably, in solving various boundary value problems. This work is aimed at overcoming the said difficulties associated with the conventional Fourier series.

It is known that a continuous function f(x) defined on the interval [−π, π] can always be expanded into a Fourier series

where the expansion coefficients are calculated from

and

The Fourier series, (1.1), reduces to

if f(x) is an odd function;

and to

if f(x) is an even function.

The convergence of the Fourier series, (1.1), is well understood through the following theorems.

THEOREM 1. If

ty and to

Proof. Pages 75-78 of Ref. [

THEOREM 2. For any absolutely integrable function

Proof. Pages 70-71 of Ref. [

THEOREM 3. Let

Proof. Pages 84, 130, and 131 of Ref. [

As a matter of fact, (1.7) can be replaced by more explicit expressions [

and

where

The aforementioned convergence theorems are established based on the condition that f(x) is a periodic function of period 2π. It is known that the Fourier series of an analytic 2π-periodic function can actually converge at an exponential rate [

Assume, for example, that

where a_{m} and b_{m} are the Fourier coefficients of

the Fourier coefficients of an absolutely integrable function tend to zero as

and

If

which recovers the convergence rate for a continuous 2π-periodic function. Unfortunately, the condition,

In recognizing this slow convergence problem, the subtraction of polynomials has been developed to remove the Gibbs phenomenon with

where

The polynomials can be easily constructed, for example, using the Lanczos’s system of polynomials:

and

Lanczos polynomials of even (odd) degrees are obviously even (odd) functions. It should be noted that Lanczos polynomials are closely related to Bernoulli polynomials which are also widely used in the methods of polynomial subtraction.

The first few Lanczos polynomials can be explicitly expressed as

For complete Fourier expansion of

For the sine expansion of

For the cosine expansion of

Assume that ^{n}^{−1} continuous on [−π, π] and its n-th derivative is absolutely integrable. Then the corrected function ^{K} continuity for K ≤ n − 1 in (1.12); b) C^{2K+1} continuity for 2K + 1 < n in (1.13) and c) C^{2K+2} continuity for 2K + 2 < n in (1.14).

By recognizing the slower convergence of sine series than its cosine counterpart, a modified Fourier series was proposed as [

If _{m} and b_{m}, in (1.15) will both decay like

For a sufficiently smooth function

where

It is known that expansion coefficients a_{m} decay like

To accelerate the convergence and maintain a close similarity to classical Fourier series, an alternative trigonometric expansion of

where

and coefficients b_{p} are to be determined as described below.

THEOREM 4. Let ^{n−}^{1}continuity on the interval [0, π] and its n-th derivative is absolutely integrable (the n-th derivative may not exist at certain points). If n ³ 2, then the Fourier coefficient a_{m}, as defined in (2.4), decays at a polynomial rate as

provided that

and

Proof. By integrating by part, we have

Denote

for sufficiently large m.

Substracting (2.9) from (2.8) leads to

The first two terms in (2.10) vanish if

and

In order to have a unique and smallest set of coefficients, b_{p}, we set Q = P in (2.11) and (2.12), or equivalently, in (2.6) and (2.7). The convergence estimate, (2.5), becomes evident from (2.10) according to Theorem 2. W

Remark. If

or, more explicitly,

Alternatively, (2.4) can be expressed as

where

Equations (2.6) and (2.7) can be rewritten in matrix form as

where

and

in which

and

Determination of the coefficients, B_{1} and B_{2}, involves the inversion of a Vandermonde-like matrix

which is always invertable if x_{k} ¹ x_{j} for j ¹ k.

Consider a polynomial of degree 2P − 1

Then it is obvious that

where δ_{ij} is Kronecker’s symbol.

According to (2.28), matrix C = [c_{ik}] is actually the inverse of matrix X.

To find an explicit expression for matrix C, let

where

and

Thus, we have

Comparing (2.32) with (2.27) leads to

or

In light of (2.34), the coefficients b_{p} (

By making use of (2.21), the first few coefficients, for example, are readily found as:

and

for P=1;

and

for P = 2;

and

for P = 3.

EXAMPLE 1. Consider function

or

where

Under the current framework, this function can be expanded as:

where

for P = 1;

where

for P = 2;

where

for P = 3.

A graphic display of the results, (2.48), (2.49), (2.51), (2.53) and (2.55), is given in

Similarly,

where b_{p} are the expansion coefficients to be determined, and

THEOREM 5. Let ^{n−}^{1} continuity on the interval [0, π] and the n-th derivative is absolutely integrable (the n-th derivative may not exist at certain points). Then for

provided that

and

Proof. By integrating by part, we have

The first two terms in (3.6) will both vanish if

and

In order to have a unique and smallest set of coefficients, b_{p}, we set Q = P in (3.7) and (3.8), or equivalently, in (3.4) and (3.5). The convergence estimate, (3.3), then becomes evident according to Theorem 2. W

Remark. If

which can be further written in a shaper form as

The expansion coefficients, a_{m}, can be alternatively expressed as

where

Actually,

We can rewrite (3.4) and (3.5) in matrix form as

where

and

Following the same procedures as described earlier, coefficients b_{p} can be obtained from

Using this formula, the first several coefficients are easily determined as:

and

for P = 1;

and

for P = 2;

and

for P = 3.

EXAMPLE 2. Consider function

where

In the context of the current framework, this function can be expressed as

where

for P = 1;

where

for P = 2.

Let

where a_{m }and b_{m} are the expansion coefficients to be calculated from

and

THEOREM 6. Let ^{n−}^{1} continuity on the interval [−π, π] and the n-th derivative is absolutely integrable (the n-th derivative may not exist at certain points). Then the Fourier coefficients of

and

provided that

and

Proof. Function

and

Since

and

(4.10) and (4.11) can be rewritten as (4.6) and (4.7), respectively.

Similarly, relationship (4.5) is readily obtained from applying Theorem 5 to the odd function h(x) on interval [0, π] by recognizing that

and

The expansion coefficients of g(x) are determined from

Similarly, the expansion coefficients of h(x) are determined from

The even (odd) extension of

(

Alternatively, (4.2) and (4.3) can be expressed as

and

where

The coefficients

and

EXAMPLE 3. Consider function

By setting P = 0 and Q = 1 in (4.1), we have

and

where

It is seen from (4.25) that the sine series now converges at a rate of m^{−3} which is faster than m^{2} for its cosine counterpart. If desired, the convergence of the series expansion in the form of (4.1) can be further accelerated by setting P = Q = 1. Accordingly, in addition to (4.23), we have

and

where

The series expansion given in (4.27) will converge at a rate of m^{−3} in comparison with m^{−2} for that in (4.24).

COROLLARY 1. Let ^{n−}^{1}continuity on the interval [0, π] and the n-th derivative is absolutely integrable (the n-th derivative may not exist at certain points). Assume n ³ 2. Then

and

Provided that

and

where

Proof. For

and

or, alternatively, (5.3) and (5.4).

Then expansion (5.1) follows immediately from (2.3) in view that

Since

COROLLARY 2. Let ^{n−}^{1} continuity on the interval [0, π] and the n-th derivative is absolutely integrable (the n-th derivative may not exist at certain points). Then for

and

provided that A_{m }and

Proof. By (5.5) and (5.6), we have

Thus, (5.7) and (5.8) become obvious from (3.1) and (3.3), respectively. W

COROLLARY 3. Let ^{n−}^{1} continuity on the interval [−π, π] and the n-th derivative is absolutely integrable (the n-th derivative may not exist at certain points. Then for

and

and

provided that

and

where

Since Corollary 3 is obvious from Theorem 6 and Corollaries 1 and 2, its proof will not be given here.

Notice that in (5.1)

where

and

Remark. In Corollary 1, (5.1) can be alternatively written as

where

and

Similarly, (5.7) in Corollary 2 can be written as

where

and

And (5.9) in Corollary 3 as

where

and

Alternative Fourier series expansions have been presented in an effort of better representing a sufficiently smooth function in a compact interval. The series expansions can take various forms, resulting in different rates of convergence. When one of the series expansions, for example, is used to solve a boundary value problem, its convergence rate needs to be compatible with the smoothness of the solution “physically” dictated by the problem. Thus, there may exist the best form for any given problem. Among other important applications, the new Fourier series will potentially lead to a new path for solving differential equations and boundary value problems.

Li, W.L. (2016) Alternative Fourier Series Expansions with Accelerated Convergence. Applied Mathematics, 7, 1824-1845. http://dx.doi.org/10.4236/am.2016.715152