相关论文: Menshov' "adjustment theorem" with respect to gene…

In this paper, we generalize Bochkarev's theorem, which states that for any uniformly bounded biorthonormal system $\Phi$, there exists a Lebesgue integrable function whose Fourier series with respect to the system $\Phi$ diverges on a set…

We prove a refined version of the celebrated Lusin type theorem for gradients by Alberti, stating that any Borel vector field $f$ coincides with the gradient of a $C^1$ function $g$, outside a set $E$ of arbitrarily small Lebesgue measure.…

A result of H. Steinhaus states that any positive Lebesgue measurable set has a property that its difference set contains an open interval around the origin. Y. V. Mospan proved that this result is the characterization of absolutely…

Let $n \in \mathbb{Z}_{\geq 3}$ be given. We prove Lebesgue-almost everywhere pointwise inversion formulae for the Siegel transforms in the geometry of numbers. These inversion formulae are quite general; for instance, they are valid for…

Let $n, m, k$ be positive integers with $k=n-m+1$. We establish an abstract Morse-Sard-type theorem which allows us to deduce, on the one hand, a previous result of De Pascale's for Sobolev $W^{k,p}_{\textrm{loc}}(\mathbb{R}^n,…

In the present paper, we study a set that can be treated as a generalised set of subsums for a geometric series. This object was discovered independently in various mathematical aspects. For instance, it is closely related to various…

It is well-known that a strict analogue of the Birkhoff Ergodic Theorem in infinite ergodic theory is trivial; it states that for any infinite-measure-preserving ergodic system the Birkhoff average of every integrable function is almost…

Minkowski's Theorem asserts that every centered measure on the sphere which is not concentrated on a great subsphere is the surface area measure of some convex body, and, moreover, the surface area measure determines a convex body uniquely.…

This article studies the rearrangement problem for Fourier series introduced by P.L. Ulyanov, who conjectured that every continuous function on the torus admits a rearrangement of its Fourier coefficients such that the rearranged partial…

Indicator functions mentioned in the title are constructed on an arbitrary nondiscrete locally compact Abelian group of finite dimension. Moreover, they can be obtained by small perturbation from any indicator function fixed beforehand. In…

This paper shows that finitely additive measures occur naturally in very general Divergence Theorems. The main results are two such theorems. The first proves the existence of pure normal measures for sets of finite perime- ter, which yield…

We first present a modern simple proof of the classical ergodic Birkhoff's theorem and Bourgain's homogeneous bilinear ergodic theorem. This proof used the simple fact that the shift map on integers has a simple Lebesgue spectrum. As a…

We prove that the classical Menshov-Rademacher, Orlicz, and Tandori theorems remain true for orthogonal series given in the direct integrals of measurable collections of Hilbert spaces. In particular, these theorems are true for the spaces…

The famous Carleson-Hunt theorem has been in focus of interest for a long time. This theorem concerns convergence almost everywhere of Fourier series of $f\in L_p$ functions for $1<p\leq \infty.$ Kolmogorov constructed a function $f\in L_1$…

We study global convex solutions of the Monge-Amp\`ere equation \[ \det D^2 u = \mu \quad \text{in } \mathbb{R}^n, \] where $\mu \not\equiv 0$ is a nonnegative locally finite periodic Borel measure on $\mathbb{R}^n$. We prove a…

It is shown that the approximating functions used to define the Bochner integral can be formed using geometrically nice sets, such as balls, from a differentiation basis. Moreover, every appropriate sum of this form will be within a…

We develope a new and general notion of parametric measure models and statistical models on an arbitrary sample space $\Omega$ which does not assume that all measures of the model have the same null sets. This is given by a diffferentiable…

Using the notion of higher-order Fourier dimension introduced in \cite{M2} (which was a sort of psuedorandomness condition stemming from the Gowers norms of Additive Combinatorics), we prove a maximal theorem and corresponding…

Given a finite Borel measure $\mu$ on R n and basic semi-algebraic sets $\Omega$\_i $\subset$ R n , i = 1,. .. , p, we provide a systematic numerical scheme to approximate as closely as desired $\mu$(\cup\_i $\Omega$\_i), when all moments…

This is an attempt of a comprehensive survey of the results in which estimates of the norms of linear means of multiple Fourier series, the Lebesgue constants, are obtained by means of estimating the Fourier transform of a function…