# Bounding Multivariate Trigonometric Polynomials with Applications to Filter Bank Design

arXiv: https://arxiv.org/abs/1802.09588 · HTML: https://mlsanity.com/paper/1802.09588/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Feb 2018.

## Built on by (most cited first, 5 of 5)

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- [Exponentially improved efficient machine learning for quantum many-body states with provable guarantees](https://mlsanity.com/paper/2304.04353/) (2023)
- [Spectral Norm of Convolutional Layers with Circular and Zero Paddings](https://mlsanity.com/paper/2402.00240/) (2024)
- [LUCID: Learning-Enabled Uncertainty-Aware Certification of Stochastic Dynamical Systems](https://mlsanity.com/paper/2512.11750/) (2025)
- [Kernel-Based Learning of Safety Barriers](https://mlsanity.com/paper/2601.12002/) (2026)

## Abstract (arXiv)

The extremal values of multivariate trigonometric polynomials are of interest in fields ranging from control theory to filter design, but finding the extremal values of such a polynomial is generally NP-Hard. In this paper, we develop simple and efficiently computable estimates of the extremal values of a multivariate trigonometric polynomial directly from its samples. We provide an upper bound on the modulus of a complex trigonometric polynomial, and develop upper and lower bounds for real trigonometric polynomials. For a univarite polynomial, these bounds are tighter than existing bounds, and the extension to multivariate polynomials is new. As an application, the lower bound provides a sufficient condition to certify global positivity of a real trigonometric polynomial. We use this condition to motivate a new algorithm for multi-dimensional, multirate, perfect reconstruction filter bank design. We demonstrate our algorithm by designing a 2D perfect reconstruction filter bank.