1.4 Uniform Convergence

1.4 Uniform Convergence

• The semicircle is centered at the origin.

• The convergence is treated at individual points of an interval.
• Uniform convergence in an interval is a stronger kind of convergence.

• There are two important facts.
• If a series converges uniformly in a period interval, it must converge to a continuous function.

• It is uniform.

• Figure 10 shows graphs of a "sawtooth" function and partial sums of its Fourier series.
• The convergence is uniform.

• One way to prove uniform convergence is to look at the coef ficients.

• If f is periodic and continuous and has a sectionally continuousderivative, the Fourier series corresponding to f converges uniformly to the real axis.

• There is no such difficulty with the even periodic extension.

• Determine if the following functions converge uniformly or not.

• The convergence is not different.

• Determine if the following functions converge uniformly.

• If you want to decide whether the convergence of the associate Fourier series is uniform, use Theorem 1.