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Convergence of Fourier Series
To this point, we have discussed the calculation of Fourier coefficients,
but this does not guarantee it will converge nor have we discussed under
which conditions a sequence of coefficients may converge. If the set of functions
which are continuous at all but a finite number of points forms a vector space
and if the trigonometric functions for a basis of this vector space, then the
answer is clear. But are there other conditions?
Question: Under what conditions of
does the Fourier series converge?
Answer: In 1829, Dirichlet found that if the following conditions are satisfied
- for some period ,
- is bounded, that is, for some finite ,
- has a finite number of extreme points (maxima and minima) over ,
- has a finite number of jump discontinuities over ,
it follows that the Fourier series representation converges to
- at any point where the function is continuous, and
- (the average of the left and right limits) at any point where the function has a jump discontinuity.
The functions and
defined on the interval
do not satisfy the conditions because the first
function is unbounded on that interval and the second, though bounded, has an infinite number
of maxima and minima.
Remark: The Dirichlet conditions are on sufficient, but not necessary. That means that:
- If the conditions are satisfied, then the Fourier series representation exists,
- If the conditions are not satisfied, then we cannot say anything about the convergence of the
Fourier series. That is, even if the conditions are not satisfied, the Fourier series may
still converge to the target function.
In practice and with almost 200 years of experience, it has been observed that the Dirichlet criteria are all that we
need for applications.
Remark: Even when the Fourier series representation of a function converges, it is necessary to know the rate
of convergence, since this is an indication of how many terms must be taken in order to obtain an accurate
approximation. Now recall that the Fourier series is
where the sine and cosine functions oscillate between -1 and 1 for all th terms, and therefore the convergence of the
series will depend strictly on the coefficients
decreasing to zero as
and at the rate at which they decrease.
Examples of Convergence
Let us go back to the example in the lecture on half-range expansions. One example we looked at
was the function defined on
. Figures 1, 2, and 3 show the coefficients for each of these expansions.
Figure 1. The coefficients for the full-range expansion of the function on dropping according to
Figure 2. The coefficients for the even expansion of the function on dropping according to
(every second term is zero).
Figure 2. The coefficients for the odd expansion of the function on dropping according to
We would say that the terms are dropping according to
for the full-range and odd expansion but
for the even expansion. Consequently, we would require significantly fewer
terms if we used the even cosine expansion.
Recall that is undefined but
for higher powers in the denominator, the infinite sum is ever closer to 1.
Remark: notice that the even expansion is continuous while the full-range and odd expansions
are discontinuous. Long experience using Fourier series has taught us:
- If the function has at least one discontinuity, the rate of convergence
of the coefficients will be ,
- If the function is continuous but the derivative is discontinuous, the
rate of convergence will be , and
- If the function and all of its derivatives up to
are continuous but
is discontinuous, then the
rate of convergence will be .
Note to instructor: do not teach how Fourier series may be used to calculate infinite sums.