Asymptotics of Fourier coefficients of power-type functions

Question

I would like to understand the asymptotic behaviour of the Fourier coefficients of power type functions $f(t) = |t|^{-\alpha} 1_{[-\pi, \pi]} \qquad 0 < \alpha<1.$ I suppose this is a classic result that I am supposed to know which can be found in many books, but I do not know where to start reading. Can you give me a hint please?

Answer 1

$$ \begin{aligned} &\int_0^\pi t^\beta e^{iyt}\frac{dt}{t}dt=y^{-\beta}\int_0^{\pi y} t^{\beta} e^{it}\frac{dt}{t}= \cr &=y^{-\beta}e^{i\pi\beta/2}\left[\int_0^{\pi y} t^\beta e^{-t}\frac{dt}{t}+O(t^{\beta-1})\right]= y^{-\beta}e^{i\pi\beta/2}[\Gamma(\beta)+o(1)]. \end{aligned} $$

Now just take the real part. If you want more terms, apply Laplace on the arc used for moving the interval of integration to the imaginary axis.

Answer 2

This is a cosine transform of $t^{-a} \mathbb{1}[0, 1],$ which can be evaluated explicitly using the trusty mathematica, which gives: $ \frac{t^{a+1} \, _1F_2\left(\frac{a}{2}+\frac{1}{2};\frac{1}{2},\frac{a}{2}+\frac{3}{2};-\frac{1}{4} k^2 t^2\right)}{a+1} $ If you want the asymptotic of the above expression in $k$ (the transform variable), you can use the mathematica command Series[your_favorite_expression, {k, Infinity, 10}] (10 gives you the first ten terms in the power series, feel free to use your favorite integer). If you use 1 instead of 10 (for ease of typesetting), you get this. (sorry, easier to use mathurl than do line breaks by hand).