Assuming that a given function can be represented by a Fourier series, the spectral coefficients in Eq. (4.4-1) are given by
The inner product notation is used here, as defined by The fundamental justification for (4.4-2) is the Fourier Series Theorem and Sell, 1982); its basis in functional analysis is beyond the scope of this book. A heuristic derivation of is obtained by forming inner products of with both sides of (4.4- and then using the orthonormality of the basis functions, to give
Equation (4.4-2), without the inner product notation, was used indirectly in Example 4.2-2. That is, the constants and defined by Eqs. (4.2-21) and (4.2-22) are the spectral coefficients for the functions respectively, the temperatures speci- fied along two of the boundaries.
It is noteworthy that Eq. (4.4-1) is analogous to the more familiar
of arbitrary vector as a sum of components, each multiplied by a corresponding base vector. The spectral coefficients correspond to the components and the basis func- tions are analogous to the base vectors. Just as the base vectors for a given coordinate system are (usually) defined so that they are mutually and normalized to unit magnitude, the basis functions in Eq. (4.4-1) are orthonormal. The inner product opera- tion Eq. (4.4-3) is like forming the scalar or dot product of a base vector with some other vector. The scalar product selects the component of the second vector correspond- ing to the particular base vector, just as isolates the corresponding
coefficient. The main difference is that the number of base vectors needed in a physical problem is ordinarily just three, whereas the number of basis functions in (4.4-1) is infinite. The fact that there is infinite number of basis functions in any set tempts one that it would not hurt to omit one. However, just as it is impossible to represent three-dimensional vector using only and (while omitting it is im- possible to accurately represent arbitrary function using an incomplete set of basis
A very readable introduction to the concept of orthogonal bases in finite- (vector) space versus infinite-dimensional (function) space is given by (1 963).
Fourier series representations of functions of two or more variables follow
immediately from (4.4-2), by simply treating the additional variables as parameters. For example, expansion for a function y) can be written as
Equation (4.4-4) is the same as the expansion for the temperature in Example 4.2-2, whereas restates the definition of the of y), given previously by It is seen that the transformed temperature, is a spectral coefficient. Thus, the strategy in the method was to compute the spectral coefficients for the two-dimensional temperature field, which in depended on the spectral coefficients for certain boundary data and B,).
Convergence
To discuss the convergence of Fourier series we return to functions of one variable. Let represent the partial sum involving the first N terms the series for or
The extent to which these partial sums converge as N is increased may be described both in a local and in an average sense. An important result pertaining to local or convergence applies when both f (x) f '(x) are piecewise continuous on the interval [a, b] (Hildebrand, 1976; Greenberg, 1988). If that is true, then
=
at each point in the open interval (a, b) for In Eq. (4.4-7), and f represent the right- and left-hand limits of the function, respectively, evaluated at Thus, at points in (a, b) where the function is continuous, the series converges to at a jump discontinuity, convergence is to the arithmetic mean of the limits. A subtle but important distinction is that the convergence described by (4.4-
7) is guaranteed at all interior points, including those arbitrarily close to the endpoints, but not at the themselves.
The overall or average convergence properties of a Fourier series are best de- scribed using the concept of a norm. The of a function written as is
, defined as
which involves the inner product of the function with itself. This inner product provides a measure of the magnitude of the function, with respect to a specific interval [a, b]
weighting function It is analogous to forming the scalar (dot) product of a vector with itself, and then taking the square root to evaluate the magnitude of the vector. For Fourier series it is found that
embodiments of those processes) tend to smooth the temperature or concentra- tion profile, usually ensuring that the field variable its first derivatives are continu- ous phase. Discontinuities in the field variable may exist at internal interfaces or boundaries, but such discontinuities ordinarily consist of finite jumps. Accordingly, almost conduction or diffusion problem will involve only piecewise continuous functions. A more general classification of functions which can be repre- sented (4.4-1) has been developed using linear operator theory [see and Sell (1982) and
Calculation of Spectral Coefficients
Assuming that a given function can be represented by a Fourier series, the spectral coefficients in Eq. (4.4-1) are given by
The inner product notation is used here, as defined by The fundamental justification for (4.4-2) is the Fourier Series Theorem and Sell, 1982); its basis in functional analysis is beyond the scope of this book. A heuristic derivation of is obtained by forming inner products of with both sides of (4.4- and then using the orthonormality of the basis functions, to give
Equation (4.4-2), without the inner product notation, was used indirectly in Example 4.2-2. That is, the constants and defined by Eqs. (4.2-21) and (4.2-22) are the spectral coefficients for the functions respectively, the temperatures speci- fied along two of the boundaries.
It is noteworthy that Eq. (4.4-1) is analogous to the more familiar
of arbitrary vector as a sum of components, each multiplied by a corresponding base vector. The spectral coefficients correspond to the components and the basis func- tions are analogous to the base vectors. Just as the base vectors for a given coordinate system are (usually) defined so that they are mutually and normalized to unit magnitude, the basis functions in Eq. (4.4-1) are orthonormal. The inner product opera- tion Eq. (4.4-3) is like forming the scalar or dot product of a base vector with some other vector. The scalar product selects the component of the second vector correspond- ing to the particular base vector, just as isolates the corresponding
coefficient. The main difference is that the number of base vectors needed in a physical problem is ordinarily just three, whereas the number of basis functions in (4.4-1) is infinite. The fact that there is infinite number of basis functions in any set tempts one that it would not hurt to omit one. However, just as it is impossible to represent three-dimensional vector using only and (while omitting it is im- possible to accurately represent arbitrary function using an incomplete set of basis
A very readable introduction to the concept of orthogonal bases in finite- (vector) space versus infinite-dimensional (function) space is given by (1 963).
Fourier series representations of functions of two or more variables follow
immediately from (4.4-2), by simply treating the additional variables as parameters. For example, expansion for a function y) can be written as
Equation (4.4-4) is the same as the expansion for the temperature in Example 4.2-2, whereas restates the definition of the of y), given previously by It is seen that the transformed temperature, is a spectral coefficient. Thus, the strategy in the method was to compute the spectral coefficients for the two-dimensional temperature field, which in depended on the spectral coefficients for certain boundary data and B,).
Convergence
To discuss the convergence of Fourier series we return to functions of one variable. Let represent the partial sum involving the first N terms the series for or
The extent to which these partial sums converge as N is increased may be described both in a local and in an average sense. An important result pertaining to local or convergence applies when both f (x) f '(x) are piecewise continuous on the interval [a, b] (Hildebrand, 1976; Greenberg, 1988). If that is true, then
=
at each point in the open interval (a, b) for In Eq. (4.4-7), and f represent the right- and left-hand limits of the function, respectively, evaluated at Thus, at points in (a, b) where the function is continuous, the series converges to at a jump discontinuity, convergence is to the arithmetic mean of the limits. A subtle but important distinction is that the convergence described by (4.4-
7) is guaranteed at all interior points, including those arbitrarily close to the endpoints, but not at the themselves.
The overall or average convergence properties of a Fourier series are best de- scribed using the concept of a norm. The of a function written as is
, defined as
which involves the inner product of the function with itself. This inner product provides a measure of the magnitude of the function, with respect to a specific interval [a, b]
weighting function It is analogous to forming the scalar (dot) product of a vector with itself, and then taking the square root to evaluate the magnitude of the vector. For Fourier series it is found that
In words, the of the involved in approximating with vanishes as
This is referred to as convergence in the mean (Churchill, 1963).
An interesting result which is related to Eq. (4.4-9) is that, for a fixed number of terms choosing the coefficients in Fourier series in the manner prescribed by Eq. (4.4-2) has the effect of minimizing the norm of the error (Churchill, 1963). In other words, a Fourier series containing a finite number of terms provides the best "least squares" approximation to the function Another result, which concerns the spectral coefficients themselves, is that
= 0.
if an infinite series involving a set of basis functions has coefficients which do not exhibit this behavior, then it is not a legitimate Fourier series.
Following are some examples of Fourier series representations for simple func- tions, which illustrate typical convergence behavior and other features of such series. The selection of examples was influenced by the fact that, when constructing a Fourier series for boundary or initial data as part of an solution, the choice of basis func- tions is ordinarily dictated by other parts of the problem. Thus, the basis functions which must be used to expand some function are not necessarily those which yield the simplest Fourier series for that function. All of these examples employ
basis functions chosen from Table 4- 1, with the interval
Example Fourier-Sine Series for = 1 With the method it is often necessary to represent a constant by a Fourier series. Suppose it is desired to represent f ( x ) = 1 using the sines given as case I in Table From (4.4-2), the are given by
sin It follows that the desired sine series is
sin
1
=
Although it may seem quite improbable that the right-hand side equals unity, this result is
of this series are shown in Fig. based on partial sums involving either 4 or 20 terms. expected, is a much better approximation to = 1 than is However, no matter
contributions in remarkable paper submitted by Fourier in 1807 to the de France were
in the first footnote of Chapter What was not mentioned is that, after review by La- Monge, and the paper was rejected. Although the other reviewers favored acceptance, La- adamantly to the idea that trigonometric series could be used to represent arbitrary Thus, any skepticism which the reader might have when first encountering an identity like excusable. Although known to other prominent mathematicians, Fourier's results were not
published until the appearance of book in The 1807 manuscript was rediscovered many his death and, with biographical information, published after a delay of 165 years I
X
4-3. Fourier-sine series representations of based on 4 or 20 terms in The exact
function is also shown.
how many terms are used, this particular series at the endpoints, because the basis functions vanish there and does not. This illustrates how increasing the number of terms gives conver- gence in the mean and local convergence at interior points, including points progressively closer to the ends of the interval, but does not necessarily help at the endpoints themselves. Incidentally, notice that only the odd-numbered "harmonics" (odd values of n) contribute to the sum in Eq. (4.4-12). This occurs whenever the function is symmetric about the midpoint of the interval,
as is the case for any constant.
If the cosines in case IV of Table 4-1 had been chosen to represent = I, the problem would have been exceptionally easy, because it would have been sufficient to set = and =
for This underscores the point that if happens to have the same functional as one
of the basis functions, a one-term representation is exact and an infinite series is not required. That situation is analogous to a vector which parallels one of the coordinate axes and which therefore has a single component.
Example 4.4-2 Fourier-Sine Series Again choosing the sines in case I of Table
4- 1, the coefficients for f (x) = - are
and the complete series is given as
sin
.
this case there is no symmetry about x = and all harmonics Plots of this series are in Fig. The behavior near x = is similar to that in the previous example. but in this the series is able to give the exact value of zero at =
In words, the of the involved in approximating with vanishes as
This is referred to as convergence in the mean (Churchill, 1963).
An interesting result which is related to Eq. (4.4-9) is that, for a fixed number of terms choosing the coefficients in Fourier series in the manner prescribed by Eq. (4.4-2) has the effect of minimizing the norm of the error (Churchill, 1963). In other words, a Fourier series containing a finite number of terms provides the best "least squares" approximation to the function Another result, which concerns the spectral coefficients themselves, is that
= 0.
if an infinite series involving a set of basis functions has coefficients which do not exhibit this behavior, then it is not a legitimate Fourier series.
Following are some examples of Fourier series representations for simple func- tions, which illustrate typical convergence behavior and other features of such series. The selection of examples was influenced by the fact that, when constructing a Fourier series for boundary or initial data as part of an solution, the choice of basis func- tions is ordinarily dictated by other parts of the problem. Thus, the basis functions which must be used to expand some function are not necessarily those which yield the simplest Fourier series for that function. All of these examples employ
basis functions chosen from Table 4- 1, with the interval
Example Fourier-Sine Series for = 1 With the method it is often necessary to represent a constant by a Fourier series. Suppose it is desired to represent f ( x ) = 1 using the sines given as case I in Table From (4.4-2), the are given by
sin It follows that the desired sine series is
sin
1
=
Although it may seem quite improbable that the right-hand side equals unity, this result is of this series are shown in Fig. based on partial sums involving either 4 or 20 terms. expected, is a much better approximation to = 1 than is However, no matter
contributions in remarkable paper submitted by Fourier in 1807 to the de France were
in the first footnote of Chapter What was not mentioned is that, after review by La- Monge, and the paper was rejected. Although the other reviewers favored acceptance, La- adamantly to the idea that trigonometric series could be used to represent arbitrary Thus, any skepticism which the reader might have when first encountering an identity like excusable. Although known to other prominent mathematicians, Fourier's results were not
published until the appearance of book in The 1807 manuscript was rediscovered many his death and, with biographical information, published after a delay of 165 years
I
X
4-3. Fourier-sine series representations of based on 4 or 20 terms in The exact
function is also shown.
how many terms are used, this particular series at the endpoints, because the basis functions vanish there and does not. This illustrates how increasing the number of terms gives conver- gence in the mean and local convergence at interior points, including points progressively closer to the ends of the interval, but does not necessarily help at the endpoints themselves. Incidentally, notice that only the odd-numbered "harmonics" (odd values of n) contribute to the sum in Eq. (4.4-12). This occurs whenever the function is symmetric about the midpoint of the interval,
as is the case for any constant.
If the cosines in case IV of Table 4-1 had been chosen to represent = I, the problem would have been exceptionally easy, because it would have been sufficient to set = and =
for This underscores the point that if happens to have the same functional as one
of the basis functions, a one-term representation is exact and an infinite series is not required. That situation is analogous to a vector which parallels one of the coordinate axes and which therefore has a single component.
Example 4.4-2 Fourier-Sine Series Again choosing the sines in case I of Table
4- 1, the coefficients for f (x) = - are
and the complete series is given as
sin
.
this case there is no symmetry about x = and all harmonics Plots of this series are in Fig. The behavior near x = is similar to that in the previous example. but in this the series is able to give the exact value of zero at =
0
0 0.2 0.4 0.6 0.8
x
Fourier-sine series representations of based on 4 or 20 terms in (4.4-14). The exact is also shown.
4.4-3 Fourier-Cosine for a Step Function Consider the step function defined
which will be represented now using the cosines in case of Table 4-1. A discontinuous function like this is handled by performing the integration for the inner product in a manner. In
this case the piece for vanishes, leaving
where the minus sign applies for n =0, 1, 4, ... and the plus sign is for n = 2. 6 , 7. ... . A convenient representation of the series is obtained by using four sums involving
a new index:
from this series are plotted in Fig. 4-5. Notice that (4.4-17) yields 8 when the upper limit for m is 1, and 40 the upper limit is 9. Once again, the approximation fails at one of the boundaries no matter how many terms are used. In accordance
series evidently converges to at the arithmetic mean of the right- and left-hand limits at the The actual computed values are and
0.493.
of an Arbitrary Functions 151
Figure 4-5. Fourier-cosine series representations of a step function, based on 8 or 40 terms in The exact function is also shown.
En Fig. 4-5 there is a distinct overshoot evident near the discontinuity even for a series with as many as 40 terms. An unfortunate property of Fourier series is that the amplitude
of the overshoot near any discontinuity eventually fails to decay further as more and more
are added. This is called the Gibbs phenomenon; see, for example, Morse and Feshbach
With sines or cosines the overshoot reaches an asymptotic limit of about 9% of the size of the
step. Similar overshoots are also seen in Figs. and 4-5 near the boundaries where wise convergence fails. Convergence in the mean is not violated, because the overshoots are confined to progressively thinner regions as the number of terms is increased.