Accurate estimates of deviations of spline approximations to classes of differentiable functions

We derive the approximation on [0, 1] of functionsf(x) by interpolating spline-functions s_r(f; x) of degree 2r+1 and defect r+1 (r=1, 2,...). Exact estimates for ¦f(x)–s_r(f; x) ¦ and f(x)–s_r(f; x)|_c on the class W^mH for m=1, r=1, 2, ..., and m=2, 3, r=1 for the case of convex (t),are derived.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 483–494, May, 1971.     

Integral representations in Hardy classes and best approximations in certain functional spaces

For the case of a simply connected domain in the plane one proves necessary and sufficient conditions for the representation of functions of the Hardy class H¹ by an integral with respect to the harmonic measure of its boundary values. A theorem is given, characterizing the rate of decrease of the best polynomial approximations of an entire function in Hardy classes by the order and the type of this function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 342–347, March, 1991.     

Integral representations in hardy classes and best approximations in certain functional spaces

For the case of a simply connected domain in the plane one proves necessary and sufficient conditions for the representation of functions of the Hardy class H¹ by an integral with respect to the harmonic measure of its boundary values. A theorem is given, characterizing the rate of decrease of the best polynomial approximations of an entire function in Hardy classes by the order and the type of this function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 342–347, March, 1991.     

Linear approximations for differential equations in Banach spaces and Lyapunov stability

Doklady Mathematics -     

The conditions for the imbedding of the classes H k,R ω \tilde H_{k,R}^\omega

We find the necessary and sufficient conditions for the imbeddings in terms of the majorants and (R=L₂,C₂;l>k).Translated from Matematicheskie Zametki, Vol. 13, No. 2, pp. 169–178, February, 1973.     

Hardy-Sobolev spaces and their multipliers

In this paper, some properties of Hardy-Sobolev spaces are obtained. The multipliers on these spaces are defined, and our results show that the multiplier algebra is more complex than that on the classical Hardy spaces. In addition, the spectrum theorem is obtained for some special multiplier.     

Derivations for even part of finite-dimensional modular Lie superalgebra \tilde \Omega

Y. Z. Zhang and Q. C. Zhang [J. Algebra, 2009, 321: 3601?C3619] constructed a new family of finite-dimensional modular Lie superalgebra $\tilde \Omega $ . Let ?? denote the even part of the Lie superalgebra $\tilde \Omega $ .We first give the generator sets of the Lie algebra ??. Then, we reduce the derivation of ?? to a certain form. With the reduced derivation and a torus of ??, we determine the derivation algebra of ??.     

Pointwise multipliers of Orlicz spaces

We show that the result on multipliers of Orlicz spaces holds in general. Namely, under the assumption that three Young functions Φ₁, Φ₂ and Φ, generating corresponding Orlicz spaces, satisfy the estimate ${\Phi^{-1}(u) \leq C \Phi_1^{-1}(u)\, \Phi_2^{-1}(u)}We show that the result on multipliers of Orlicz spaces holds in general. Namely, under the assumption that three Young functions Φ₁, Φ₂ and Φ, generating corresponding Orlicz spaces, satisfy the estimate F^-1(u) ￡ C F₁^-1(u) F₂^-1(u){\Phi^{-1}(u) \leq C \Phi_1^{-1}(u)\, \Phi_2^{-1}(u)} for all u > 0, we prove that if the pointwise product xy belongs to L ^Φ(μ) for all y ? L^F₁(m){y \in L^{\Phi_1}(\mu)}, then x ? L^F₂(m){x \in L^{\Phi_2}(\mu)}. The result with some restrictions either on Young functions or on the measure μ was proved by Maligranda and Persson (Indag. Math. 51 (1989), 323–338). Our result holds for any collection of three Young functions satisfying the above estimate and for an arbitrary complete σ-finite measure μ.     

A characterization of multipliers in the Hedberg-Netrusov spaces

We expose a criterion of the membership of a function in the space of pointwise multipliers between the Hedberg-Netrusov spaces. The criterion is stated in terms of a pair of discrete weighted inequalities. Connections of the criterion with the problem of rectifiability of Lipschitz surfaces are established.     

Fourier multipliers of Lipschitz spaces

Let G denote an infinite, compact, metrizable, 0-dimensional, Abelian group. The following are characterized: (i) the multipliers from one Lipschitz space Lip(α, p; G) to another Lipschitz space Lip(β, q; G) for 0 < α < β < ∞ and 1 ? p, q ? ∞; and (ii) the multipliers from Lip(α, p; G) to Lip(β, q; G) for 0 < β ? α < ∞ and 1 < q ? 2 ? p < ∞. Two special cases of (i), namely the case q = ∞ and the case p = 1, were obtained by the authors in an earlier publication (1981). A. Zygmund (J. Math. Mech.8 (1959), 889–895) and T. Mizuhara (Tôhoku Math. J.24 (1972), 263–268) have characterized the multipliers of certain Lipschitz spaces defined on the circle group.     

Mixed equilibrium problems with relaxed \alpha -monotone mapping in Banach spaces

Using the KKM mapping technique, we obtain existence results for the mixed equilibrium problem (under the relaxed \(\alpha \)-monotonicity assumption on the bi-function) in a reflexive Banach space. Several examples are given to show that our definition of \(\alpha \)-monotonicity and, hence, our results are more general than that in the existing literature.     

Laws of the iterated logarithm for $$\tilde \rho$$-mixing random variables with normal distribution

Consider a \(\tilde \rho\)-mixing sequence of identically distributed random variables with the underlying distribution in the domain of attraction of the normal distribution. This paper proves that law of the iterated logarithm holds for \(\tilde \rho\)-mixing sequences of random variables. Our results generalize and improve Theorems 1.2–1.3 of Qi and Cheng (1996) from the i.i.d. case to \(\tilde \rho\)-mixing sequences.     

On large deviations for approximations of SDEs

 We establish sufficient conditions under which the rate function for the Euler approximation scheme for a solution of an SDE is close to that for an exact solution.