On the order of approximation to functions ofH p (R) (0<p≦1) by certain means of Fourier integrals

H ^P (R ₊ ² ) — R ₊ ² ={zC: Imz>0} ^p (R) — H ^p (R ₊ ² ). P ^k (f,x) — ë- — ,W ^k (f,x) — — R ^k, (f,x) — f H (R) (. §1,1)–3)); _k (, f) _p — - . , fH ^p (R) 0<p1,kN; (1+)^–1<p1, 0<<,kN.     

Subcategories of fixed points of mutations in root categories with type _(n-1)

For any n 3, let R(n) denote the root category of finite-dimensional nilpotent representations of cyclic quiver with n vertices. In the present paper, we prove that R(n-1) is triangle-equivalent to the subcategory of fixed points of certain left (or right) mutation in R(n). As an application, it is shown that the affine Kac-Moody algebra of type n-2 is isomorphic to a Lie subalgebra of the Kac-Moody algebra of type n-1.     

Existence of weak solutions of 2-D Euler equations with initial vorticityω 0∈L(log+ L) α (α>0)

It is proved that there exist global weak solutions of 2-D Euler equations inR ² under the assumption that the initial vorticity belongs to a kind of wider spaces,L ¹L(log⁺ L) (>0), which are Orlicz spaces containing spacesL ^p L ¹,L(log⁺ L) L (>1/2) and so on. This result improves on that of [2], [4], [11]. Moreover, these solutions are obtained by vanishing the viscosity term of Navier-Stokes equations.     

A homogeneous Hilbert problem with discontinuous coefficients and two-side curling at infinity point of order 1/2 ≤ ρ < 1

We study a homogeneous Riemann??Hilbert boundary-value problem in the upper half of the complex plane with a countable set of coefficient discontinuities and two-side curling at infinity. We obtain a general solution in the case when the problem index has a power singularity of order ??, 1/2 ?? ?? < 1, and study the solvability conditions.     

On the reduction (mod 1) of completely monotone functions (0, ∞)

Summary The Euler-Maclaurin summation formula and its harmonic analysis (Poisson) are applied to the case of functions which are completely monotone on an open half-line. What thus results is a curious class of Fourier series, which can be determined explicitly and which represent completely monotone functions on the first half of the period. A by-product is the complete monotony (on the first half-period) of the Bernoulli functions, whether the index is integral or fractional.     

The moduli of certain families of curves and the range of f(ζ0) in the class of univalent functions with real coefficients

By simultaneously considering two moduli problems for pairs of homotopic classes of curves a complete solution is obtained of the problem of the range of functions of the class S_R, where S_R is the class of functions in S with real coefficients, at a fixed point ₀ of the disk ¦¦<1, and min¦f(₀¦ in the class S_Ris also found. Partial results in this problem were obtained earlier by J. Jenkins and V. V. Chernikov.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 139, pp. 156–167, 1984.     

Supplement to the Pál type (0; 0, 1) lacunary interpolation

Letx ₁, …,x _n be givenn distinct positive nodal points which generate the polynomial $$\omega _n (x) = \prod\limits_{i = 1}^n {(x - x_i )} .$$ Letx*₁, …,x* _n?1 be the roots of the derivativeω′ _n (x) and putx ₀=0. In this paper, the following theorem is proved: Ify ₀, …,y _n andy′₁, …,y′ _n?1 are arbitrary real numbers, then there exists a unique polynomialP _2n?1(x) of degree 2n?1 having the following interpolation properties: $$P_{2n - 1} (x_j ) = y_j (j = 0,...,n),$$ , $$P_{2n - 1}^\prime (x_j^* ) = y_j^\prime (j = 1,...,n - 1).$$ . This result gives the theoretical completion of the original Pál type interpolation process, since it ensures uniqueness without assuming any additional condition.     

Trace forms of abelian extensions of number fields of type (1, 0)

This article is concerned with describing certain bilinear forms associated with finite abelian extensions N|K of an algebraic number field K. These abelian trace forms are described up to Witt equivalence, that is, they are described as elements in the Witt ring W(K). When the base field K has exactly one dyadic prime and no real embeddings, it is shown that the Witt class of every abelian trace form over K is a product of Witt classes of five specified types.     

Benford’s law and distribution functions of sequences in (0, 1)

Applying the theory of distribution functions of sequences x _n ∈ [0, 1], n = 1, 2, ..., we find a functional equation for distribution functions of a sequence x _n and show that the satisfaction of this functional equation for a sequence x _n is equivalent to the fact that the sequence x _n to satisfies the strong Benford law. Examples of distribution functions of sequences satisfying the functional equation are given with an application to the strong Benford law in different bases. Several direct consequences from uniform distribution theory are shown for the strong Benford law.     

The best polynomial approximations of entire transcendental functions with generalized growth order in Banach spaces \mathcal{E}_p^{\prime}(G) and {\mathcal{E}_p}(G) , p ? 1

A theorem of the Hadamard type for entire transcendental functions f, which have a generalized ??-order of growth ?? _?? (f), has been obtained. This theorem connects the values $ \widetilde{M}\left( {f,r} \right)\;\left( {r > 1} \right) $ and the coefficients a _n (f) $ \left( {n \in {\mathbb{Z}_{+} }} \right) $ of the expansion of f in Faber series in a finite domain D whose boundary ?? belongs to the Al??per class. This result is the extension of a result obtained by M. N. Sheremeta onto a simply connected domain. The necessary and sufficient conditions for an analytic function $ f \in \mathcal{E}_p^{\prime}(G) $ or $ f \in {\mathcal{E}_p}(G)\;\left( {1 \leqslant p \leqslant \infty } \right) $ to be entire transcendental with a generalized ??-order of growth ?? _?? (f) are obtained. These conditions include the best polynomial approximations of the function f and determine the rate of their convergence to zero, as the degree of polynomials increases.     

Weak type (1, 1) of some operators for the Laplacian with drift

Let \(v = (v_1, \ldots , v_n)\) be a vector in \(\mathbb {R}^n {\setminus } \{ 0 \}\). Consider the Laplacian on \(\mathbb {R}^n\) with drift \(\Delta _{v} = \sum _{i = 1}^n \Big ( \frac{\partial ^2}{\partial x_i^2} + 2 v_i \frac{\partial }{\partial x_i} \Big )\) and the measure \(d\mu (x) = e^{2 \langle v, x \rangle } dx\), with respect to which \(\Delta _{v}\) is self-adjoint. Let d and \(\nabla \) denote the Euclidean distance and the gradient operator on \(\mathbb {R}^n\). Consider the space \((\mathbb {R}^n, d, d\mu )\), which has the property of exponential volume growth. We obtain weak type (1, 1) for the Riesz transform \(\nabla (- \Delta _{v} )^{-\frac{1}{2}}\) and for the heat maximal operator, with respect to \(d\mu \). Further, we prove that the uncentered Hardy–Littlewood maximal operator is bounded on \(L^p\) for \(1 < p \le +\infty \) but not of weak type (1, 1) if \(n \ge 2\).     

The growth of solutions of f″+ e-zf′+ Q(z)f - 0 where the order (Q)＝ I

This paper investigates the growth of solutions of the equation f" + e-zf′ + Q(z)f＝ 0 wherethe order (Q)＝ 1. When Q(z) ＝ h(z)ebz, h(z) is nonzero polynomial, b ≠ -1 is a complex constant, every solution of the above equation has infinite order and the hyper-order 1. We improve the results of M. Frei, M.Ozawa, G. Gundersen and J. K. Langley.     

( B0−$\bar B^0 $) Mixing in the next-to-leading order of the 1/ mb-expansion

We present results of calculating the matrix elements of the (B⁰−[`(B)]⁰\bar B^0 )-mixing operators in the next-to-leading order of the expansion in 1/m _b .