Lower bounds for the merit factors of trigonometric polynomials from Littlewood classes.
The authors give shorter and more direct proofs of certain results by J. E. Littlewood published in the 60's. These proofs also lead to explicit constants. The following theorem proved by them is typical: If \(p\) is a trigonometric polynomial of degree at most \(n\) with real coefficients that satisfies the conditions \[ \Biggl(\int_{\mathbb{T}} p^2(t)\,dt\Biggr)^{1/2}\leq An^{1/2}\quad\text{and}\quad \Biggl(\int_{\mathbb{T}}| p'(t)|^2\,dt\Biggr)^{1/2}\geq Bn^{3/2}, \] then \[ \Biggl({1\over 2\pi} \int_{\mathbb{T}} p^4(t)\,dt\Biggr)^{1/4}\geq \Biggl(1+{1\over 111} \Biggl({B\over A}\Biggr)^{1/2}\Biggr) \Biggl({1\over 2\pi} \int_{\mathbb{T}} p^2(t)\,dt\Biggr)^{1/2}. \] Conjugate reciprocal, unimodular polynomials on the unit circle of the complex plane are also considered.
scientific article; zbMATH DE number 2026879