Convergence of Trigonometric Fourier Series

This is a list of definitions, lemmas and theorems needed to provide convergence arguments for trigonometric Fourier series. Some proofs begin on page 3.

Definitions

  1. For any nonnegative integer k, a functionu is if every k-th order partial derivative of u exists and is continuous.
  2. For two functions f and g defined on an interval , we will define the inner product as
  3. A function f is periodic with period p if for all x.
  4. Let f be a function defined on such that The periodic extensionof f is the unique periodic function of period such that for all
  5. is called the N-thDirichlet Kernel. [This will be summed later and the sequences of kernels converges to what is called the Dirac Delta function.]
  6. A sequence of functions is said to converge pointwise toon the interval if for each fixedx in the interval,
  7. A sequence of functions is said to converge uniformly toon the interval if
  8. One-sided limits: and .
  9. A function f is piecewise continuous on if the function satisfies
  10. f is defined and continuous at all but a finite number of points of
  11. For all the limits and exist.
  12. and exist.
  13. A function is piecewise on if andare piecewise continuous on

Lemmas

  1. Bessel’s Inequality: Let be defined on and If the trigonometric Fourier coefficients exist, then This follows from the previous handout, Orthogonality and the Least Squares Approximation.
  2. Riemann-Lebesgue Lemma: Under the conditions of Bessel’s Inequality, the Fourier coefficients approach zero as This is based upon some earlier convergence results seen in Calculus in which one learns for a series of nonnegative terms, with if does not approach 0 as then does not converge. Therefore, the contrapositive holds, if converges, then as From Bessel’s Inequality, we see that when f is square integrable, the series formed by the sums of squares of the Fourier coefficients converges. Therefore, the Fourier coefficients must go to zero as n increases. This is also referred to in the last handout, Orthogonality and the Least Squares Approximation. However, an extension to absolutely integrable functions exists, which is called the Riemann-Lebesgue Lemma.
  3. Green’s Formula: Let f and g be functions on Then [Note: This is just an iteration of integration by parts.]
  4. Special Case of Green’s Formula: Let f and g be functions on and both functions satisfy the conditions and Then
  5. Lemma 1: If g is a periodic function of period and c any real number, then
  6. Lemma 2: Let f be a function on such that and Then for and and .
  7. Lemma 3: For any real such that
  8. Lemma 4: Let be on Then

Theorems

  1. Theorem 1.(Pointwise Convergence) Let f be on with Then for all x in
  2. Theorem 2.(Uniform Convergence)Let f be on withThen converges uniformly to. In particular, for all x in where
  3. Theorem 3. (Piecewise C1 – Pointwise Convergence) Let f be a piecewise function on. Then converges to the periodic extension of for all x in
  4. Theorem 4.(Piecewise C1– Uniform Convergence) Let f be a piecewise function on such that Then converges uniformly to.

Proof of Convergence

We are considering the Fourier series of

,

where the Fourier coefficients are given by

We are first interested in the pointwise convergence of the infinite series. Thus, we need to look at the partial sums for each x. Writing out the partial sums, inserting the Fourier coefficients and rearranging, we have

Here is called the N-th Dirichlet Kernel. What we seek to prove is (Lemma 4) that [Technically, we need the periodic extension of f.] So, we need to consider the Dirichlet kernel. Then pointwise convergence follows, as

Proposition:

Proof:Actually, this follows from Lemma 3. Let and multiply by to obtain:

Thus, or if

If then one needs to apply L’Hospital’s Rule:

As the Dirac delta function, on the interval.The following are some plots for and Note how a central peak grows and the values tend towards zero for nonzero x.

N=25 / N=50 / N=100

The Dirac delta function can be defined as that quantity satisfying (a) (b)

This generalized function, or distribution,also has the property:

Thus, under the appropriate conditions on f, one can show We need to prove Lemma 4 first.

Proof: Since we have that

The two terms look like the Fourier coefficients. An application of the Riemann-L:ebesgue Lemma indicates that these coefficients tend to zero as provided the functions being expanded are square integrable and the integrals above exist. The cosine integral follows, but a little work is needed for the sine integral. One can use L’Hospital’s Rule with

Now we apply Lemma 4 to get the convergence from Due to periodicity, we have

We can apply Lemma 4 providing is in z, which is true since f is and behaves well at

To prove Theorem 2 on uniform convergence, we need only combine Theorem 1 with Lemma 2. Then we have,

This gives the uniform convergence.

These Theorems can be relaxed to include piecewise functions. Lemma 4 needs to be changed for such functions to the result that by splitting the integral into integrals over, and applying a one-sided L’Hospital’s Rule. Proving uniform convergence under the conditions in Theorem 4 takes a little more effort, but it can be done.