The following technique for approximating the characteristic function of intervals in $[0,1]$ using Fourier series with coefficients having inverse quadratic decay is given in the book Diophantische Approximationen (Ergebnisse der Math., IV, 4; Berlin, 1936) by J.F.Koksma. Let us consider the characteristic function $\chi_{[0,\gamma]}$ of the interval $[0,\gamma]$ with $\gamma < 1$. In page 99 of the book, it is claimed that for every $\eta>0$ there exist continuous functions $G_1$ and $G_2$ on $[0,1]$ such that, $G_1(x) \leq \chi_{[0,\gamma]}(x) \leq G_2(x)$ and $G_1$ and $G_2$ have Fourier expansions, $G_1(t)=\gamma-\eta+ \sum\limits_{n \neq 0} A^1_n e^{2 \pi i n x}$ and $G_2(t)=\gamma+\eta+ \sum\limits_{n \neq 0} A^2_n e^{2 \pi i n x}$ such that for $i \in \{1,2\}$, the following bound is true:
$$\lvert A_n^i\rvert \leq \min \left\{ \frac{1}{\lvert n \rvert},\frac{1}{n^2 \eta}\right\}.$$
The existence of these bounding functions $G_1$ and $G_2$ with fast decaying Fourier coefficients is helpful in using in the Weyl's criterion for equidistribution, for example in bounding the fraction of times a sequence falls within the interval $[0,\gamma]$. It is claimed (in page 99) that the result follows by considering the Fourier series approximations for the following choice of functions $G_1$ and $G_2$. $G_1(x)$ is defined to be $1$ when $0 \leq x \leq \gamma$, $0$ when $\gamma+\eta \leq x \leq 1-\eta$ and linear when $1-\eta \leq x \leq 1$ or $\gamma \leq x \leq \gamma+ \eta$. Similarly, $G_2(x)$ is defined to be $1$ when $\eta \leq x \leq \gamma-\eta$, $0$ when $\gamma \leq x \leq 1$ and linear when $0 \leq x \leq \eta$ or $\gamma-\eta \leq x \leq \gamma$.
I tried to verify the claim by calculating the Fourier series of these functions using an integration tool and after simplifying some expressions involving exponential functions, I was able to show the second bound:
$$\lvert A_n^i\rvert \leq \frac{1} {n^2 \eta} $$
This implies that for $n$ such that $\lvert n \rvert \geq \frac{1}{\eta}$ we have $\lvert A_n^i\rvert \leq \frac{1}{\lvert n \rvert}$. Hence,
$$\lvert A_n^i\rvert \leq C_\eta\min \left\{ \frac{1}{\lvert n \rvert},\frac{1}{n^2 \eta}\right\}.$$
where the constant $C_\eta$ depends on $\eta$. But, I am unable to remove the constant $C_\eta$ and obtain the exact bound that was claimed in the book. I would appreciate it if anyone can answer the following questions:
- Can you demonstrate how the asymptotic result above can be improved to obtain the exact bound that was claimed in the book? If possible, I would like to see a proof that does not expand out the Fourier coefficient integrals completely but instead uses some bounding techniques to arrive at the required upper bounds on $\lvert A_n^i\rvert$.
- Using a different choice of functions $G_1$ and $G_2$ can we obtain approximations for characteristic functions of intervals in $[0,1]$ where the Fourier coefficients decay faster than the inverse quadratic speed (in $n$) given in the bounds above? Does there exist a choice of approximating functions with geometric decay for the Fourier coefficients?
I was wondering if some class of smooth functions used in Fourier analysis can give an optimal answer to the second question above.