Vol 54, No 5 (2019)

The paper is devoted to the solvability questions of the Wiener-Hopf integral equation in the case where the kernel K satisfies the conditions 0 ≤ K ∈ L₁(ℝ), \(\int_{-\infty}^{\infty} K(t)dt>1\), K(±x) ∈ C⁽³⁾(ℝ⁺), (−1)ⁿK(±x)⁽ⁿ⁾(x) ≥ 0, x ∈ ℝ⁺, n =1, 2, 3. Based on Volterra factorization of the Wiener-Hopf operator, and invoking the technique of nonlinear functional equations, we construct real-valued solutions both for homogeneous and non-homogeneous Wiener-Hopf equations, assuming that the function g is real-valued and summable, and the corresponding conditions are satisfied. The behavior at infinity of the corresponding solutions is also studied.

Different generalized Cesáro summation methods are compared with each other. Analogous of Hardy’s theorem, concerning the order of the partial sums of trigonometric Fourier series, for generalized Cesáro means are obtained.

The present paper is devoted to the study of existence and uniqueness of a solution and Ulam type stabilities for Volterra delay integro-differential equations on a finite interval. Our analysis is based on the Pachpatte’s inequality and Picard operator theory. Examples are provided to illustrate the stability results obtained in the case of a finite interval. Also, we give an example to illustrate that the Volterra delay integro-differential equations are not Ulam-Hyers stable on the infinite interval.

The paper deals with the question of convergence of multiple Fourier-Haar series with partial sums taken over homothetic copies of a given convex bounded set \(W\subset\mathbb{R}_+^n\) containing the intersection of some neighborhood of the origin with \(\mathbb{R}_+^n\). It is proved that for this type sets W with symmetric structure it is guaranteed almost everywhere convergence of Fourier-Haar series of any function from the class L(ln⁺L)ⁿ⁻¹.

We derived a convergence result for a sequential procedure known as alternating maximization (minimization) to the maximum likelihood estimator for a pretty large family of models - Generalized Linear Models. Alternating procedure for linear regression becomes to the well-known algorithm of Alternating Least Squares, because of the quadraticity of log-likelihood function L(υ). In Generalized Linear Models framework we lose quadraticity of L(υ), but still have concavity due to the fact that error-distribution is from exponential family. Concentration property makes the Taylor approximation of L(υ) up to the second order accurate and makes possible the use of alternating minimization (maximization) technique. Examples and experiments confirm convergence result followed by the discussion of the importance of initial guess.