Prove that the Fourier series of $\frac{1}{f}$ is absolutely convergent

Question

I have a problem:

Let $f$ be a continuous function on the unit circle $(\Gamma)$:

$$\Gamma=\{e^{i\theta}: \theta\in [0, 2 \pi]\}$$

Assume that $f \ne 0$ on $\Gamma$, and the Fourier series of $f$ is absolutely convergent on $\Gamma$.

Prove that the Fourier series of $\dfrac{1}{f}$ is absolutely convergent on $\Gamma$.

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• I've tried to use the definition of $\sum_{n=0}^\infty a_n$ is absolutely convergent:

A real or complex series $\sum_{n=0}^\infty a_n$ is said to converge absolutely if $\sum_{n=0}^\infty \left|a_n\right| = L$ for some real number $L$.

• And use the definition of the Fourier series

So let the Fourier series of $f$ be given by: $$f(x) = \sum_{n=-\infty}^{\infty} a_{n}e^{in \theta}$$

We want to show that $$\sum_{n=-\infty}^{\infty} |a_{n}e^{in \theta}| \to f(\theta)< +\infty$$

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But I still have no solution :( . Can anyone help me!

Any help will be appreciated! Thanks!

Answer 1

The standard proof ever since Gelfand uses commutative Banach agebras, but this requires a certain amount of preparations. A short proof was given by Newman in 1975. The paper is only two pages long, but still too long to summarize. Anyway, the paper is accessible freely at http://www.ams.org/journals/proc/1975-048-01/S0002-9939-1975-0365002-8/S0002-9939-1975-0365002-8.pdf