I am reading 'A Panorama of Harmonic Analysis" by S. Krantz, and on page 67, he writes; ($D_N$ here denotes the Dirichlet kernel)
The point of the last paragraph is worth belaboring. For each $N$, let $\phi_N(t)$ be the function that equals +1 when $D_N(t)\geq0$ and equals -1 when $D_N(t)<0$. Of course $\phi_N(t)$ is discontinuous. But now let $\psi_N(t)$ be a continuous function, bounded in absolute vale by 1, which agrees with $\phi_N$ except in a very small interval about each point where $\phi_N$ changes sign. Integrate $D_N$ against $\psi_N$. The calculation alluded at the start of the section then shows that the value of the integral is about $c\cdot\log N$, even though $\psi_N$ has supremum norm 1. The uniform boundedness principle now tells us that convergence for partial summation in norm fails dramatically for continuous functions on the circle group.
I'm having trouble understanding that last sentence. What does he mean by 'fails dramatically'? How is the uniform boundedness principle used here?
Let us denote the circle group by $\mathbb{T}$ and let $S_N f(t)$ be the $N$th partial sum of the Fourier series of a given function $f:\mathbb{T}\rightarrow\mathbb{C}$.
Does he mean that the family of continuous functions (on the circle group) which $S_N f(t)$ does not converges to $f$ in uniform norm, is dense in $C(\mathbb{T})$?
Please enlighten me!
I do not have the book of Krantz. Anyway, I think that the point is the following (you can find a detailed explanation in Rudin, R&CA, pp. 100-103).
Consider the linear functionals $$ L_N \colon C(\mathbb{T}) \to \mathbb{R}, \qquad L_N f := S_N f (0). $$ (Here we consider the point $x=0$, but any other point does the job.)
The computation you refer to tells us that $$ \| L_N \| = \| D_N \|_1 \sim c \cdot \log N \to +\infty. $$ But then, from the uniform boundedness principle, it follows that $$ \sup_N |S_N f (0) | = +\infty $$ for every $f$ in some dense $G_\delta$-set in $C(\mathbb{T})$.