An example of an almost greedy uniformly bounded orthonormal basis for $L_p([0,1])$

Selection of subsystems of random variables equivalent in distribution to the Rademacher system

We present necessary and sufficient conditions on systems of random variables for them to possess a lacunary subsystem equivalent in distribution to the Rademacher system on the segment [0,1]. In particular, every uniformly bounded orthonormal system has this property. Furthermore, an arbitrary finite uniformly bounded orthonormal set of N functions contains a subset of "logarithmic" density equivalent in distribution to the corresponding set of Rademacher functions, with a constant independent of N. A connection between the tail distribution and the L_p-norms of polynomials with respect to sy…

Sidonicity and variants of Kaczmarz's problem

We prove that a uniformly bounded system of orthonormal functions satisfying the $ψ_2$ condition: (1) must contain a Sidon subsystem of proportional size, (2) must satisfy the Rademacher-Sidon property, and (3) must have its 5-fold tensor satisfy the Sidon property. On the other hand, we construct a uniformly bounded orthonormal system that satisfies the $ψ_2$ condition but which is not Sidon. These problems are variants of Kaczmarz's Scottish book problem (problem 130) which, in its original formulation, was answered negatively by Rudin. A corollary of our argument is a new, elementary proof …

divergent Fourier series in function spaces near $L^1[0;1]$

In this paper we generalize Bochkariev's theorem, which states that for any uniformly bounded orthonormal system $Φ$, there exists a Lebesgue integrable function such that the Fourier series of it with respect to system $Φ$ diverge on the set of positive measure. We characterize the class of variable exponent Lebesgue spaces $L^{p(\cdot)}[0;1]$, $1<p(x)<\infty$ a.e. on [0;1], such that above mentioned Bochkarev's theorem is valid.

非局部凸洛伦兹序列空间中平均投影的有界性及其在基理论中的应用

Boundedness of the averaging projections in nonlocally convex Lorentz sequence spaces and applications to basis theory

研究对称Schauder基在准巴拿赫空间中平均投影的有界性，发现非局部凸情况下该性质不成立，但某些加权洛伦兹序列空间的canonical基具有统一有界平均投影，推翻了局部凸性由有界平均投影决定的假设，并在非局部凸空间中得到新的基结构结论。