Fourier series convergence question from big Rudin.

Question

I am working on some problems from the 3rd edition of Rudin's "Real and Complex Analysis" and I'm stumped on proving the following part from question #19 of chapter 5.

Suppose $\lambda_n/\log n \to 0$ as $n \to \infty$. Prove that there exists an $f \in C(T)$ such that the sequence $\lbrace s_n(f;0)/\lambda_n \rbrace$ is unbounded.

Here $s_n(f;0)$ denotes the n'th partial sum of the Fourier series of $f$ at $t=0$, $\hspace{1mm} T$ is the unit circle, and $\lambda_n$ is a sequence.

Help would be greatly appreciated; thank you in advance!

Answer 1

Follow the hint given in the question: use the argument in Sec. 5.11, use the Banach-Steinhaus Theorem in exactly the same way with the change of the linear functionals to ${\Lambda _n}f = \frac{1}{{{\lambda _n}}}{s_n}(f;0)$. The proof is almost word for word there. Again the hint suggest a better estimate for the 1-norm of the Dirichlet kernel D_n. In the argument on page 162 in Rudin's book, replace D_n by D_n / λ_n and give a better estimate in line 5-6 there.

You will get

$\frac{4}{{{\pi ^2}|{\lambda _n}|}}\sum\limits_{k = 1}^n {\frac{1}{k}} \ge \frac{4}{{{\pi ^2}|{\lambda _n}|}}(\ln (n) + {\gamma}) = \frac{4}{{{\pi ^2}}}(\frac{{\ln (n)}}{{|{\lambda _n}|}} + \frac{{{\gamma}}}{{|{\lambda _n}|}}) \to \infty $

since you are given $\frac{{{\lambda _n}}}{{\ln (n)}} \to 0$ .

Here $\gamma$ is the Euler Mascheroni constant.

Follow the remaining argument there to invoke Banach-Steinhaus Theorem to give the conclusion.