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Is there any intuitive definition for the class of periodic functions that have only non-negative Fourier coefficients? Or at least (relatively weak) sufficient conditions for a periodic function to have non-negative Fourier coefficients?

I am particularly interested in describing a class of even functions with period $2\pi$ and zero mean value that satisfy the condition above. By writing the expression for the Fourier coefficients of such function $f$, it becomes obvious that it has non-negative Fourier coefficients if and only if

$$\int_0^\pi f(x) \cos(nx)\,dx \geq 0\quad\forall n \geq 1.$$

I would like to indentify sufficient conditions for $f$ to satisfy the inequality above. An example of a function that satisfies this inequality is the triangular wave (with maximum positive amplitude at $x=0$, with zero mean value and period $2\pi$).

Thank you in advance!

1 Answers 1

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A sufficient condition is that $f$ can be written as a convolution of some $2\pi$-periodic function $g$ with $g(-x)$, that is $$ f(x) = \int_{-\pi}^{\pi} g(x+t)g(t)\,dt $$ First of all, such $f$ is even because $$ f(-x) = \int_{-\pi}^{\pi} g(-x+t)g(t)\,dt = \int_{-\pi-s}^{\pi-x} g(s)g(x+s)\,ds = \int_{-\pi}^{\pi}g(s)g(x+s)\,ds = f(x) $$ Secondly, the cosine coefficients are nonnegative because $$ \int_{-\pi}^{\pi} f(x)\cos n x\,dx = \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} g(x+t)g(t) \cos n x\,dx \,dt $$ where after expanding $$ \cos nx = \cos (n(x+t) - nt) = \cos n(x+t) \cos nt + \sin n(x+t)\sin nt $$ and using Fubini's theorem we arrive at the sum of $$ \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} g(x+t)g(t) \cos n(x+t)\cos nt \,dx \,dt = \left(\int_{-\pi}^{\pi} g(x)\cos nx \,dx \right)^2 $$ and $$ \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} g(x+t)g(t) \sin n(x+t)\sin nt \,dx \,dt = \left(\int_{-\pi}^{\pi} g(x)\sin nx \,dx \right)^2 $$

Remarks

  1. The above computation is cleaner in terms of complex exponential Fourier series, where it becomes "Fourier coefficients of a convolution are the product of Fourier coefficients of the terms".
  2. There are no sufficient condition that we can immediately perceive in terms of $f$ itself. Indeed, a tiny bump somewhere, undetectable with any kind of integral or pointwise inequalities, can make some high-frequency coefficient negative.
  3. The triangular wave is a special case, with $g$ being a rectangular wave.
  4. Bochner's theorem gives a necessary and sufficient condition for having nonnegative Fourier coefficients, but it's not so easy to verify.