Volume 16, Issue 3

The sufficient conditions are established for the existence of solutions for a class of boundary value problems for fractional differential equations involving the Caputo fractional derivative.

The behavior of generalized Cesàro (𝐶, α 𝑛 )-means ( α 𝑛 ∈ (–1, 0), 𝑛 = 1, 2, . . .) of conjugate trigonometric Fourier series of 𝐻 𝑤 classes in the space of continuous functions is studied.

In [Yoshino, Proc. Amer. Math. Soc. 95: 571–572, 1985] the author proved that for a 𝑀-hyponormal operator 𝐴* and for a dominant operator 𝐵, 𝐶𝐴 = 𝐵𝐶 implies 𝐶𝐴* = 𝐵*𝐶. In the case where 𝐴* and 𝐵 are normal, this result is known as the Fuglede–Putnam theorem. In this paper, we will extend this result to the case in which 𝐴 is an injective (𝑝, 𝑘)-quasihyponormal operator and 𝐵* is a dominant operator. We also show that the same result remains valid for (𝑝, 𝑘)-quasihyponormal and log-hyponormal operators.

For any multiplicative system of bounded type and any set of measure zero there exists a bounded measurable function whose Fourier series with respect to this system diverges on this set.

We study the uniqueness problems on entire or meromorphic functions concerning differential polynomials that share one slowly growing meromorphic function. Furthermore, we greatly improve and generalize some former results obtained by Fang and Hong, Lin and Yi, Xiong and Lin.

We construct a uniformly bounded orthonormal almost greedy basis for the variable exponent Lebesgue spaces 𝐿 𝑝(·) ([0, 1]), 1 < 𝑝 – ≤ 𝑝 + ≤ 2 (or 2 ≤ 𝑝 – ≤ 𝑝 + < ∞), when the diadic Hardy–Littlewood maximal operator is bounded on these spaces.

We study a certain integral modification of the well known Szasz–Mirakyan operators with a weight function of a general Baskakov basis function. We establish some direct results on a simultaneous approximation for these new operators.

The main aim of this paper is to prove that the (𝐶, α )-means of quadratic partial sums of double Walsh–Kaczmarz–Fourier series are of weak type (1, 1) and of type (𝑝, 𝑝) for all 1 < 𝑝 ≤ ∞ (0 < α < 1). Moreover, these (𝐶, α )-means converge to 𝑓 almost everywhere for any integrable function 𝑓.

The main aim of this paper is to prove that the partial sums of double Vilenkin–Fourier series do not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of 𝐿ln + 𝐿, the set of functions whose quadratic partial sums of double Vilenkin–Fourier series converge in measure is of first Baire category.

In the paper, in particular, it is proved that the 𝑠-dimensional trigonometric Fourier series of the 2 π -periodic continuous on [– π , π ] 𝑠 function 𝑓 is uniformly strong summable to the function 𝑓 exponentially in the power . Moreover, it is proved that this result is best possible.

Analogues of two Hardy–Littlewood theorems are proved for a multiplicative Fourier transform. A Szasz type condition for a multiplicative Fourier transform is given and its nonimprovability is proved. Besides, an analogue of Ul'yanov's theorem on the uniform convergence of a trigonometric series and an analogue of Konyuškov–Stechkin's embedding theorem are obtained by means of a Nikol'skii type inequality of various metrics.

It is proved that the Hilbert transform defined on a finite interval is bounded in the grand Lebesgue space if and only if 𝑤 satisfies the Muckenhoupt condition 𝐴 𝑝 .

We prove the necessary conditions for the integrability of the Fourier transform. The result is a generalization, on one hand, of the well known necessary condition for absolutely convergent Fourier series and, on the other hand, of an earlier multidimensional result of Trigub.

For any holomorphic function 𝑓 on the unit polydisk we consider its restriction to the diagonal, i.e., the function in the unit disc 𝔻 ⊂ ℂ defined by Diag 𝑓(𝑧) = 𝑓(𝑧, . . . , 𝑧) and prove that the diagonal map Diag maps the space of the polydisk onto the space of the unit disk.

A derivation basis is introduced in a compact zero-dimensional metric space 𝑋. A Henstock–Kurzweil type integral with respect to this basis is defined and used to represent the so-called quasi-measure on 𝑋.

In the present paper, the connection between a T-direction and Borel direction of algebroid functions is investigated. Two algebroid functions (not meromorphic) are obtained to prove that T-directions and Borel directions are two different classes of singular directions for algebroid functions. The existence of an algebroid function (not meromorphic) with a given Borel direction is proved.