Bound on maximum of a fourier transform

Question

Could someone show me why the following relation holds?

$ \max_{\lambda} \vert f(\lambda)\vert \leq \sum_{-m}^m\vert t_k\vert $

where

$ f(\lambda) $ is the fourier transform of the sequence $ \{t_k \} $.

Thank you.

Dario

Answer 1

I'm supposing that $\lambda \in \mathbb{R}$. For complex $\lambda$, the inequality would be in general false.

So let's just compute for arbitrary real $\lambda$:

$$\lvert f(\lambda)\rvert = \left\lvert \sum_{k=-m}^m t_k e^{ik\lambda} \right\rvert \leqslant \sum_{k=-m}^m \lvert t_k\rvert\cdot \left\lvert e^{ik\lambda}\right\rvert = \sum_{k=-m}^m \lvert t_k\rvert.$$

The first equality is by definition of $f$, the inequality is the triangle inequality, and the next equality uses $\lvert e^{ix}\rvert = 1$ for $x\in\mathbb{R}$.