Vol 54, No 4 (2019)

In this paper the concept of interassociativity via hyperidentities of associativity is extended and describe the set of semigroups which are {i, j}-interassociative to the free semigroup and the free commutative semigroup, where i, j = 1, 2, 3.

In this paper we study the Dirichlet problem in the half-space for regular hypoelliptic equations. Applying a special integral representation, we construct approximate solutions for this problem and thereby prove correct solvability of the problem.

In 1987 Harris proved-among others that for each 1 ≤ p < 2 there exists a two-dimensional function f ∈ L_p such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space L_p(II²) (1 ≤ p < 2) with subsequence of triangular partial means \(S_{2^A}^\Delta(f)\) of the double Walsh-Fourier series convergent in measure on II² is of first Baire category in L_p(II²). We also prove that for each function f ∈ L₂(II²) a.e. convergence \(S_{a(n)}^\Delta (f) \rightarrow f\) holds, where a(n) is a lacunary sequence of positive integers.

Let {X(t), t ∈ ℝ} be a centered real-valued stationary Gaussian process with spectral density f. The paper considers a question concerning asymptotic distribution of tapered Toeplitz type quadratic functional \(Q_T^h\) of the process X(t), generated by an integrable even function g and a taper function h. Sufficient conditions in terms of functions f, g and h ensuring central limit theorems for standard normalized quadratic functionals \(Q_T^h\) are obtained, extending the results of Ginovyan and Sahakyan (Probability Theory and Related Fields 138, 551–579, 2007) to the tapered case and sharpening the results of Ginovyan and Sahakyan (Electronic Journal of Statistics 13, 255–283, 2019) for the Gaussian case.