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Lecture 27

Lecture 26 | Lecture 28

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 tex:$$f(t)$$ does the Fourier series converge?

Answer: In 1829, Dirichlet found that if the following conditions are satisfied

  • tex:$$f(t) = f(t + T)$$ for some period tex:$$T$$,
  • tex:$$f(t)$$ is bounded, that is, tex:$$|f(t)| \le M$$ for some finite tex:$$M$$,
  • tex:$$f(t)$$ has a finite number of extreme points (maxima and minima) over tex:$$T$$,
  • tex:$$f(t)$$ has a finite number of jump discontinuities over tex:$$T$$,

it follows that the Fourier series representation converges to

  • tex:$$f(t)$$ at any point tex:$$t$$ where the function is continuous, and
  • tex:$$\frac{1}{2}\left(f(t^-) + f(t^+)\right)$$ (the average of the left and right limits) at any point tex:$$t$$ where the function has a jump discontinuity.

Invalid Examples

The functions tex:$$f(t) = \frac{1}{t + 1}$$ and tex:$$f(t) = \sin\left(\frac{1}{t + 1}\right)$$ defined on the interval tex:$$(-1, 1)$$ 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

tex:$$f(t) = \frac{c_0}{2} + \sum_{n = 1}^\infty c_n \cos(n\omega t) + \sum_{n = 1}^\infty s_n \sin(n\omega t)$$

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 tex:$$c_n$$ and tex:$$s_n$$ decreasing to zero as tex:$$n \rightarrow \infty$$ 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 tex:$$f(t) = t$$ defined on tex:$$[0, 4]$$. 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 tex:$$f(t) = t$$ on tex:$$[0, 4]$$ dropping according to tex:$$-\frac{4}{n\pi}$$.

Figure 2. The coefficients for the even expansion of the function tex:$$f(t) = t$$ on tex:$$[0, 4]$$ dropping according to tex:$$-\left(\frac{4}{n\pi})^2$$ (every second term is zero).

Figure 2. The coefficients for the odd expansion of the function tex:$$f(t) = t$$ on tex:$$[0, 4]$$ dropping according to tex:$$(-1)^n\frac{8}{n\pi}$$.

We would say that the terms are dropping according to tex:$$O(n^{-1})$$ for the full-range and odd expansion but tex:$$O(n^{-2})$$ for the even expansion. Consequently, we would require significantly fewer terms if we used the even cosine expansion.

Recall that tex:$$\sum_{n = 0}^\infty \frac{1}{n}$$ is undefined but tex:$$\sum_{n = 0}^\infty \frac{1}{n^2} = \frac{\pi^2}{6} \approx 1.645$$ and that 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 tex:$$O(n^{-1})$$,
  • If the function is continuous but the derivative is discontinuous, the rate of convergence will be tex:$$O(n^{-2})$$, and
  • If the function and all of its derivatives up to tex:$$k$$ are continuous but tex:$$f^{(k)}(t)$$ is discontinuous, then the rate of convergence will be tex:$$O(n^{-(k + 2)})$$.

Note to instructor: do not teach how Fourier series may be used to calculate infinite sums.