Volume 34, Issue 3

Abstract. We consider operators defined by the Hadamard product of two harmonic or analytic functions. Estimates for norms of such operators are obtained for the cases of Bergman spaces and Hardy spaces of analytic and harmonic functions in the unit disk. In some cases, the obtained estimates are sharp.

Abstract. In this paper, sequence spaces X(r,s,t;B(m))$X(r, s, t;B^{(m)})$ for X∈{l∞,c,c0}$X\in \lbrace l_\infty , c, c_0\rbrace $ are introduced by combining the generalized means and the m -th order generalized difference operator B(m)(u,v)$B^{(m)}(u, v)$. It is shown that these spaces are complete normed linear spaces and the spaces c0(r,s,t;B(m))$c_0(r, s, t; B^{(m)})$, c(r,s,t;B(m))$c(r, s, t;B^{(m)})$ have Schauder bases. Furthermore, the α-, β-, γ-duals of these spaces are computed. The necessary and sufficient conditions for some matrix transformations from X(r,s,t;B(m))$X(r, s, t;B^{(m)})$ to Y , where X,Y∈{l∞,c,c0}$X, Y \in \lbrace l_\infty , c, c_0\rbrace $, are also obtained. Finally, some classes of compact operators on the spaces X(r,s,t;B(m))$X(r, s, t;B^{(m)})$ for X∈{l∞,c,c0}$X\in \lbrace l_\infty , c, c_0\rbrace $ are characterized by using the Hausdorff measure of noncompactness.

Abstract. Let ℳα$\mathcal {M}_{\alpha }$ be the class of moduli of continuity defined by Leindler in [Studia Sci. Math. Hungar. 14 (1979), 431–439], and Hωα(Ω¯)$H^{\omega _{\alpha }}(\overline{\Omega }) $ be the generalized Hölder class of functions on the closure of the regular hexagon Ω, where 0<α≤1$0&lt;\alpha \le 1$ and ωα∈ℳα$\omega _{\alpha }\in \mathcal {M}_{\alpha }$. The difference f-𝒱nλ(f)$f-\mathcal {V}_n^{\lambda }(f) $ is estimated in the uniform norm ∥·∥C(Ω¯)$\Vert \cdot \Vert _{C(\overline{\Omega })}$ and in the generalized Hölder norm ∥·∥ωβ$\Vert \cdot \Vert _{\omega _\beta }$, where 𝒱nλ(f)$\mathcal {V}_n^{\lambda }(f) $ is the n th generalized de la Vallée-Poussin mean of hexagonal Fourier series of f∈Hωα(Ω¯)$f\in H^{\omega _{\alpha }}(\overline{\Omega }) $ and 0≤β<α≤1$0\le \beta &lt;\alpha \le 1$.

Abstract. In the present paper we first examine growth properties of regular Nørlund means applied to power series outside their disk of convergence. Moreover, we show that overconvergent subsequences of Nørlund means exist, assuming that the considered power series has certain gaps.

Abstract. This is a corrected version of http://dx.doi.org/10.1515/anly-2012-1238. the paper, the authors provide four alternative proofs of an explicit formula for computing Bernoulli numbers in terms of Stirling numbers of the second kind.