# Can we control the maximum norm of a continuous signal whose finitely many Fourier coefficients are fixed?

Let us denote $$C_{2\pi}$$ by the set of all $$2\pi$$-periodic continuous signals $$x:\mathbb{R}\to \mathbb{R}$$.

Fix $$n\in \mathbb{N}$$ and put $$\Lambda_n=\{y\in C_{2\pi}: \mathcal{F}(y)[k]=0 ~\operatorname{for all}~ |k|\leq n \}$$

Let $$z\in C_{2\pi}$$ such that its Fourier coefficients satisfy $$\mathcal{F}(z)[k]=0$$ for all $$|k|>n$$ and put $$\gamma_z=\displaystyle\inf_{y\in \Lambda_n} \left\{\|x\|_{\infty} : x=z+y \right\}$$

Q. Can we conclude that $$\gamma_z=0$$? If NOT, how accurately can $$\gamma_z$$ be estimated?

p.s. $$\|x\|_{\infty}$$ is just the maximum of the absolute value of $$x$$.

• I'm a bit confused about $z$ and $y$. You define $\Lambda_z$ for a signal $y$ (with Fourier coefficients vanishing for $|k|\le n$?), and you define $\gamma_z$ for yet another signal $x$, defined as the sum $y+z$. This means that $x$ has no vanishing coefficients, because, according to your definition, $y$ has non-zero coefficients for $|k|>n$, and $z$ has non-zero coefficients for $|k|\le n$. I generally like the clarity of mathematical notation, but here it seems to obfuscate the actual question. Jan 2 at 14:01
• I did some minor change in the notations.
– ABB
Jan 2 at 14:53
• The signal $x=y+z$ generally has infinitely many Fourier coefficients (up to index $n$ from $z$, and all higher ones from $y$), so how does that fit in with the title of your question? Jan 2 at 15:03

Q. Can we conclude that $$\gamma_z=0$$?

Unless I'm missing something here the answer is a clear no. You simply add two signals, one being a lowpasses and the other being high passed. The sum will have all frequencies.

If NOT, how accurately can $$\gamma_z$$ be estimated?

Assuming $$\gamma_z$$ is simply the maximum amplitude of $$x$$ (I can't fully parse your math expression here) the answer is simple enough. The maximum possible amplitude is the absolute sum of the Fourier Coefficients (more or less) which includes $$\infty$$ for periodic pulse train of dirac deltas.

To determine the actual maximum amplitude (including the phase) you just have to take the inverse Fourier Transform and calculate. Not sure whether that qualifies as "estimation".

• Yes. Finding time domain limits from frequency domain parameters generally only yields very loose bounds.
– Peter K.
Jan 2 at 15:41