Vol 214, No 2 (2016)

A method for studying the nonself-adjoint integral operators associated with differential equations of fractional order is presented. Within this method, in particular, some estimates for the eigenfunctions and eigenvalues of a boundary-value problem for a fractional oscillatory equation are obtained.

Let \( \mathbb{D} \) be an open unit disk in the complex plane. It is shown that every subspace in C(\( \mathbb{D} \)) invariant under weighted conformal shifts contains a radial eigenfunction of the corresponding invariant differential operator. This function can be expressed via the Gauss hypergeometric function and is a generalization of the spherical function on the disk \( \mathbb{D} \) which is considered as a hyperbolic plane with the corresponding Riemannian structure.

Generalized solvability of the classical boundary-value problems for analytic and quasiconformal functions in arbitrary Jordan domains with boundary data that are measurable with respect to the logarithmic capacity is established. Moreover, it is shown that the spaces of the found solutions have the infinite dimension. Finally, some applications to the boundary-value problems for A-harmonic functions are given.

The systems of rational functions {Φ_n(z)}, n ∈ ℤ; that are orthonormalized on the real axis ℝ and are defined by the fixed set of points a := {a_k}_k = 0^∞, (Im a_k > 0) and b := {b_k}_k = 1^∞, (Im b_k < 0); are considered. Some analogs of the Dirichlet kernels of the systems {Φ_n(t)}, n ∈ ℤ; on the real axis ℝ are given in a compact form, and the convergence in the spaces L_p(ℝ); p > 1; and the pointwise convergence of Fourier series on the systems {Φ_n(t)}, n ∈ ℤ; are studied under the certain restrictons on the sequences of poles of these systems. Some analogs of the classical Jordan–Dirichlet and Dini–Lipschitz criteria of convergence of Fourier series in a trigonometric system are constructed.