2019
Том 71
№ 4

All Issues

Golubov B. I.

Articles: 2
Article (English)

Fourier cosine and sine transforms and generalized Lipschitz classes in uniform metric

Golubov B. I., Volosivets S. S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 616-627

For functions $f \in L^1(\mathbb{R}_{+})$ with cosine (sine) Fourier transforms $\widehat{f}_c(\widehat{f}_s)$ in $L^1(\mathbb{R})$, we give necessary and sufficient conditions in terms of $\widehat{f}_c(\widehat{f}_s)$ for $f$ to belong to generalized Lipschitz classes $H^{\omega, m}$ and $h^{\omega, m}$. Conditions for the uniform convergence of the Fourier integral and for the existence of the Schwartz derivative are also obtained.

Article (Russian)

On Modified Strong Dyadic Integral and Derivative

Golubov B. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 628-638

For functions fL(R +), we define a modified strong dyadic integral J(f) ∈ L(R +) and a modified strong dyadic derivative D(f) ∈ L(R +). We establish a necessary and sufficient condition for the existence of the modified strong dyadic integral J(f). Under the condition \(\smallint _{R_ + }\) f(x)dx = 0, we prove the equalities J(D(f)) = f and D(J(f)) = f. We find a countable set of eigenfunctions of the operators J and D. We prove that the linear span L of this set is dense in the dyadic Hardy space H(R +). For the functions fH(R +), we define a modified uniform dyadic integral J(f) ∈ L (R +).