Notes on Fourier analysis (part 2)

Michael Liu

4 years ago

1.3. Uniform convergence

Continuity is not necessarily preserved under pointwise limits. For example, Suppose that is defined by . If , then as . If , then as . Thus, is continuous on but the pointwise limit is not (it is discontinuous at 1). On the other hand, uniform convergence of continuous functions does guarantee continuity.

Definition 3.1 Uniform convergence The series of continuous function converges uniformly on the closed interval if the following two conditions are satisfied: First, the series converges to the sum for each value in the interval; Second, given any number , there exists a such that for all the partial sums satisfy for every value in the interval.

The following theorem gives an easily applied condition for determining uniform convergence. It is especially useful in its applicaion to Fourier series.

Theorem 3.2 Weierstrass’ -Test If for all points in we have from a certain on, and the series of positive numbers converges, then the series converges uniformly on .

Proof For each value in the interval , since and converges, then by comparison test, the series converges absolutely. Therefore, the series converges absolutely for every on the interval .

Given some , then there exist an such that implies .

By Triangle inequality, we have for each in

Thus, for all ,

for every in .

In the case of Fourier series, we have the following estimates

Hence the Weierstrass’s -test will apply to a Fourier series in trigonometric form if and , and to Fourier series in exponential form if .

Another theorem that we will use for the proof of uniform convergence is the Cauchy-Schwarz Inequality.

Theorem 3.3 Cauchy-Schwarz Inequality Let and be two finite sets of real numbers. Then,

When expressed in vector form, the Cauchy-Schwarz inequality says that the dot product of two vectors is bounded by the product of their norms.

Proof Expanding out the brackets and collecting identical terms, we have

The left-hand side of the equation is greater than or equal to zero since it is a sum of the squares of real numbers. Thus,

The graph of periodic functions that are both continuous and piecewise smooth is a smooth curve except that is can have “corners” where the derivatives jump. The fundamental theorem of calculus,

applies to functions that are continuous and piecewise smooth, even though is defined at the “corners”. To see this, let to be differentiable except at point , we have

We introduce the following theorem as a preliminary step towards the main convergence theorem.

Theorem 3.4 Suppose is -periodic, continuous, and piecewise smooth. Let , , and be the Fourier coefficients of defined in (2.5) and (2.6), and let , , and be the corresponding Fourier coefficients of . Then

Proof This is a simple matter of integration by parts.

From theorem 3.4 we obtain the following results on differentiation and integration of Fourier series.

Theorem 3.5 Suppose is -periodic, continuous, and piecewise smooth, and suppose also that is piecewise smooth. If

is the Fourier series of , then is the sum of the derived series

for all at which exists. At the exceptional points has jumps, the series converges to .

Proof Since is piecewise smooth, by Theorem 2.6, it is the sum of it Fourier series at every point. By theorem 3.4, the coefficients of , , in this series are , , and respectively. Thus theorem 3.5 follows.

Theorem 3.6 Suppose is periodic and piecewise continuous, with Fourier coefficients , , , and let . If , then for all we have

where the constant term is the mean value of on :

The series on the right of (72) is the series obtained by formally integrating the Fourier series of term by term, whether the latter series actually converges or not. If , the sum of the series on the right of (72) is .

Proof is continuous and piecewise smooth, being the integral of a piecewise continuous function. Moreover, if , is periodic, for

Hence, by theorem 2.6, is the sum of its Fourier series at every . But by theorem 3.4 applied to F, the Fourier coefficients , and of are related to those of by

The formula (74) for the constant or is just the usual formula for the zeroth Fourier coefficient of . If , these arguments can be applied to the function rather than , yielding the final assertion.