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When are the Fourier coefficients of $2\pi$-periodic functions summable? More specifically, which ones of $f(x) = (\cos x)^{100}$, $g(x) = \sin(\tan x)$, and $$ h(x) = \begin{cases} x+\pi, & : x\in[-\pi,0] \\ \pi-x, & : x\in (0,\pi] \end{cases} $$

are absolutely summable?

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This is called the Wiener algebra (en.wikipedia.org/wiki/Wiener_algebra). A small amount of regularity is necessary, e.g. Holder continuous for index strictly greater than 1/2. –  Elan B. Apr 30 '12 at 3:55
    
You can't do better than that, but $(cos(x))^{100} $ is a trig polynomial, and it is easy to show that if the derivative is $\mathbb L^2$ then the original is absolutely summable. –  mike Apr 30 '12 at 12:00

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up vote 1 down vote accepted

As for summability, use Dini's theorem, or at least the one I learned in my real analysis course (I don't like Wikipedia's convoluted explanation of it). If the function is $L_1$, roughly meaning that it has a finite integral on the interval, and is Lipschitz continuous on the interval, then the partial sums converge to the function value, which hopefully is finite.

So then $f$ and $h$ have convergence everywhere (due to Lipschitz), but $g$ has continuity problems at $\pm\pi/2$, so Dini can't be applied there. At a glance, I'm quite sure convergence fails for $g$ at those spots.

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