Some trigonometric polynomials with extremely small uniform norm and their applications

We construct orthogonal trigonometric polynomials satisfying a new spectral condition and such that their $L^{1}$-norms are bounded below and the uniform norm of their partial sums has extremely small order of growth. We obtain new results that relate the uniform norm and $\mathrm{QC}$-norm on subspaces of the vector space of trigonometric polynomials.