We show that certain monotone functionals on the Hardy spaces and convex functionals on the Bergman spaces are maximized at the normalized reproducing kernels among the functions of norm 1, thus proving the contractivity conjecture of Pavlović and of Brevig, Ortega-Cerdà, Seip and Zhao and the Wehrl-type entropy conjecture for the SU(1, 1) group of Lieb and Solovej, respectively.

We prove a contractive Hardy-Littlewood type inequality for functions from H p (T), 0 < p 2 which is sharp in the first two Taylor coefficients and asymptotically at infinity.

We prove that under very mild conditions for any interpolation formula $$f(x) = \sum _{\lambda \in \Lambda } f(\lambda )a_\lambda (x) + \sum _{\mu \in M} {\hat{f}}(\mu )b_{\mu }(x)$$ f ( x ) = ∑ λ ∈ Λ f ( λ ) a λ ( x ) + ∑ μ ∈ M f ^ ( μ ) b μ ( x ) we have a lower bound for the counting functions $$n_\Lambda (R_1) + n_{M}(R_2) \ge 4R_1R_2 - C\log ^{2}(4R_1R_2)$$ n Λ ( R 1 ) + n M ( R 2 ) ≥ 4 R 1 R 2 - C log 2 ( 4 R 1 R 2 ) which very closely matches recent interpolation formulas of Radchenko and Viazovska and of Bondarenko, Radchenko and Seip.

We study the frame properties of the Gabor systemsIn particular, we prove that for Herglotz windows g such systems always form a frame for L 2 (R) if α, β > 0, αβ ≤ 1. For general rational windows g ∈ L 2 (R) we prove that G(g; α, β) is a frame for L 2 (R) if 0 < α, β, αβ < 1, αβ ∈ Q and ĝ(ξ) = 0, ξ > 0, thus confirming Daubechies conjecture for this class of functions. We also discuss some related questions, in particular sampling in shift-invariant subspaces of L 2 (R).

We prove that under very mild conditions for any interpolation formulawe have a lower bound for the counting functions nwhich very closely matches interpolation formulas from [8], [3].