We define a general Franklin system of functions on R with vanishing means, generated by an admissible sequence T. A necessary and sufficient condition on T is found for the corresponding general Franklin system of functions on R with vanishing means to be an unconditional basis in the space H1(R).

In this paper we obtain appropriate integral representations for functions from Sobolev multianisotropic spaces, and apply them to obtain embedding theorems for these spaces.

Let X be a metric measure space satisfying the doubling condition of order γ > 0. For a function f ∈ L loc (X), p > 0 and a ball B ⊂ X by I () f we denote the best approximation by constants in the space L p (B). In this paper, for functions f from Hajlasz–Sobolev classes M (X), p > 0, α > 0, we investigate the size of the set E of points for which the limit lim r→+0 I (,) () f = f*(x). exists. We prove that the complement of the set E has zero outer measure for some general class of outer measures (in particular, it has zero capacity). A sharp estimate of the Hausdorff dimension of this complement is given. Besides, it is shown that for x ∈ E $$\mathop {\lim }\limits_{r \to + 0} {\int_{B\left( {x,r} \right)} {\left| {f - f*\left( x \right)} \right|} ^q}d\mu = 0,{1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q} = {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p} - {\alpha \mathord{\left/ {\vphantom {\alpha r}} \right. \kern-\nulldelimiterspace} r}.$$ Similar results are also proved for the sets where the "means" I (,) () f converge with a specified rate.

The paper considers Bergman type operators introduced by Shields and Williams depending on a normal pair of weight functions. We prove that there exist values of parameter s for which these operators are bounded on mixed norm spaces L(p, q, s) on the unit ball in Cn.

In the present paper, sequences of real measurable functions defined on a measure space ([0, 1], µ), where µ is the Lebesgue measure, are studied. It is proved that for every sequence fn that converges to f in distribution, there exists a sequence of automorphisms Sn of ([0, 1], µ) such that fn(Sn(t)) converges to f(t) in measure. Connection with some known results is also discussed.

In the unit disc bounded by the circle T = {z, |z| = 1} we consider the Riemann boundary value problem in the weighted space L1(ρ), where $$\rho \left( t \right) = {\prod\nolimits_{k = 1}^m {\left| {t - {t_k}} \right|} ^{{\alpha _k}}}$$ , tk ∈ T, k = 1, 2,..., m, and αk, k = 1, 2,..., m are real numbers. The question of interest is to determine an analytic outside the circle T function ϕ(z), ϕ(∞) = 0 to satisfy $${\lim _{r \to 1 - 0}}||{\Phi ^ + }\left( {rt} \right) - a\left( t \right){\Phi ^ - }\left( {{r^{ - 1}}t} \right) - f\left( t \right)|{|_{{L^1}\left( {{\rho _r}} \right)}} = 0$$ , where f ∈ L1(ρ), a(t) ∈ Cδ(T), δ>0, and ρr are some continuations of function ρ inside the circle. The normal solvability of this problem is established.

Gamma-type functions satisfying the functional equation f(x+1) = g(x)f(x) and limit summability of real and complex functions were introduced by Webster (1997) and Hooshmand (2001). However, some important special functions are not limit summable, and so other types of such summability are needed. In this paper, by using Bernoulli numbers and polynomials Bn(z), we define the notions of analytic summability and analytic summand function of complex or real functions, and prove several criteria for analytic summability of holomorphic functions on an open domain D. As consequences of our results, we give some criteria for absolute convergence of the functional series $$\sum\nolimits_{n = 0}^\infty {{c_n}\sigma \left( {{Z^n}} \right)} ,where\sigma \left( {{Z^n}} \right) = {S_n}\left( z \right) = \frac{{{B_{n + 1}}\left( {z + 1} \right) - {B_{n + 1}}\left( 1 \right)}}{{n + 1}}$$ . Finally, we state some open problems.

The paper deals with a question of robustness of inferences, carried out on a continuoustime stationary process contaminated by a small trend, to this departure from stationarity.We show that a smooth periodogram approach to parameter estimation is highly robust to the presence of a small trend in themodel. The obtained result is a continuous version of that of Hede and Dai (Journal of Time Series Analysis, 17, 141-150, 1996) for discrete time processes.

In this paper we obtain sufficient conditions for the bi-harmonic differential operator A = ΔE2 + q to be separated in the space L2 (M) on a complete Riemannian manifold (M,g) with metric g, where ΔE is the magnetic Laplacian onM and q ≥ 0 is a locally square integrable function on M. Recall that, in the terminology of Everitt and Giertz, the differential operator A is said to be separated in L2 (M) if for all u ∈ L2 (M) such that Au ∈ L2 (M) we have ΔE2u ∈ L2 (M) and qu ∈ L2 (M).

The Ricci flow is an evolution equation in the space of Riemannian metrics.A solution for this equation is a curve on the manifold of Riemannian metrics. In this paper we introduce a metric on the manifold of Riemannian metrics such that the Ricci flow becomes a geodesic.We show that the Ricci solitons introduce a special slice on the manifold of Riemannian metrics.