When are Fourier coefficients monotonic?

Question

Given some sufficiently smooth function $f$ what conditions would be sufficient for its Fourier coefficients, as defined by $$ \hat{f}(n) := \int_{0}^{2\pi}\cos(nx)f(x)\ dx, \quad \text{for } n = 1,2,\ldots, $$ to be monotonic? Given the decay properties of Fourier coefficients, the monotonicity result would translate to $$ |\hat{f}(n)| \geq |\hat{f}(n+1)|, \quad n = 1,2,\ldots. $$ I haven't been able to find any literature regarding this and a result of this nature would be very interesting.

Answer 1

It suffices that $f$ be (the restriction to $[0,2\pi]$ of) a completely monotone real-valued function defined on $[0,\infty)$. Indeed, then for some finite measure $\mu$ on $[0,\infty)$ and all real $x\ge0$ we have $$f(x)=\int_0^\infty\mu(da) e^{-a x},$$ whence for natural $n$ $$\hat f(n)=\int_0^\infty\mu(da) \int_0^{2\pi}dx\,\cos(nx)e^{-a x} =\int_0^\infty\mu(da) \frac{a \left(1-e^{-2 \pi a}\right)}{a^2+n^2},$$ which is obviously decreasing in $n$ (to $0$, by dominated convergence or by the Riemann--Lebesgue lemma).

Note that, if $f(x)\equiv1$ or $f(x)\equiv x$, then $\hat f(n)=0$ for all natural $n$. So, if $f$ has the desired property, then the function $[0,2\pi]\ni x\mapsto a+bx+f(x)$ also has it for any real $a$ and $b$. Also, clearly, if $f$ has the desired property, then do does the function $$[0,2\pi]\ni x\mapsto f^-(x):=f(2\pi-x)$$ -- because $\widehat{f^-}(n)=\hat f(n)$ for all natural $n$. It follows that, if $f$ and $g$ have the desired property, then the function $$[0,2\pi]\ni x\mapsto a+bx+f(x)+g(2\pi-x)$$ also has it for any real $a$ and $b$.

Added:

1. As noted in a comment by Fedor Petrov, if $f(x)=h(\pi-x)$ for some odd function $h$ and all $x\in[0,2\pi]$, then $\hat f(n)=0$ for all natural $n$.

2. It follows from this answer by fedja that, if $$f(x)=\int_1^\infty[\mu(dp) x^p+\nu(dp)(2\pi-x)^p]<\infty$$ for some measures $\mu$ and $\nu$ on $[1,\infty)$ and all $x\in[0,2\pi]$, then $f$ has the desired property.

Answer 2

A comment on this problem would be that if $\hat f(n)$ are monotone ( here $f$ any continuous function, not necessarily odd or even, also I assume that $\hat f(n)$ is monotone not $|\hat f(n)|$ ) then one can assume that they are positive. And if Fourier coefficients are real and positive then they must be absolutely convergent, that is $\{\hat f(n)\} \in l_1$. This follows easily from property of Fejer's kernel, i.e. that it is positive operator with integral 1:

$$ \sum_k (1-|k|/n)\hat f(k)exp(ikt) = \int F(t-s)f(s) \le \sup|f|$$ so $1/2 \sum_{k \in (-n/2, n/2)} \hat f(k)) \le \sup|f|$.