We consider nonharmonic Fourier series defined in terms of arbitrarily close exponentials. Our aim is to use the positivity of quadratic polynomials in order to get new Ingham-type weighted inequalities. The proof relies on an Ingham proof technique inspired by Jaffard et al. (J Fourier Anal Appl 3:577–582, 1997). As applications, we consider families of frequencies with relevance in control approximation theory, for which we can prove the uniform (with respect to the mesh-size) controllability property of the semi-discrete model, when the spurious frequencies (the gap between them tends to zero when the mesh size goes to zero) are eliminated.

我们考虑用任意接近的指数定义的非调和傅立叶级数。我们的目标是利用二次多项式的正性来得到新的英厄姆型加权不等式。该证明依赖于受 Jaffard 等人启发的 Ingham 证明技术。（J Fourier Anal Appl 3:577–582, 1997）。作为应用，我们考虑与控制近似理论相关的频率族，为此我们可以证明半离散模型的均匀（相对于网格大小）可控性，当寄生频率（它们之间的差距趋于当网格尺寸变为零时变为零）被消除。