On multiplicators of power series in Hardy spaces

We investigate the accuracy of certain sufficient conditions for multiplicators established by Trigub. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 4, pp. 585–587, April, 1998.     

Multidimensional Bilinear Hardy Inequalities

Doklady Mathematics - A characterization of the n-dimensional bilinear Hardy inequality in weighted Lebesgue spaces is given.     

Multidimensional analogues of Bohr's theorem on power series

Generalizing the classical result of Bohr, we show that if an -variable power series converges in -circular bounded complete domain and its sum has modulus less than 1, then the sum of the maximum of the modulii of the terms is less than 1 in the homothetic domain , where . This constant is near to the best one for the domain

     

Annihilator condition does not pass to polynomials and power series

In this paper we construct a ring A which has annihilator condition (a.c.) and we show that neither $A[x]$ nor $A[[x]]$ has this property. This answers in negative a question asked by Hong, Kim, Lee and Nielsen. We also show that there is an algebra A which does not have annihilator condition while both $A[x]$ and $A[[x]]$ have this property.     

Integral analog of one generalization of the Hardy inequality and its applications

Under certain conditions on continuous functions μ, λ, a, and f, we prove the inequality

A transplantation theorem for Jacobi series in weighted Hardy spaces

A transplantation theorem for Jacobi series in weighted Hardy spaces is proved.     

Ramanujan-Fourier series and the conjecture D of Hardy and Littlewood

We give a heuristic proof of a conjecture of Hardy and Littlewood concerning the density of prime pairs to which twin primes and Sophie Germain primes are special cases. The method uses the Ramanujan-Fourier series for a modified von Mangoldt function and the Wiener-Khintchine theorem for arithmetical functions. The failing of the heuristic proof is due to the lack of justification of interchange of certain limits. Experimental evidence using computer calculations is provided for the plausibility of the result. We have also shown that our argument can be extended to the m-tuple conjecture of Hardy and Littlewood.     

On the shift semigroup on the Hardy space of Dirichlet series

We develop a Wold decomposition for the shift semigroup on the Hardy space $ \mathcal{H}^2 $ \mathcal{H}^2 of square summable Dirichlet series convergent in the half-plane $ \Re (s) > 1/2 $ \Re (s) > 1/2 . As an application we have that a shift invariant subspace of $ \mathcal{H}^2 $ \mathcal{H}^2 is unitarily equivalent to $ \mathcal{H}^2 $ \mathcal{H}^2 if and only if it has the form $ \phi \mathcal{H}^2 $ \phi \mathcal{H}^2 for some $ \mathcal{H}^2 $ \mathcal{H}^2 -inner function φ.     

Transplantation theorem for Jacobi series in weighted Hardy spaces, II

Muckenhoupt's transplantation theorem for Jacobi series in weighted L ^p spaces is extended to weighted Hardy spaces.     

Hausdorff type operators on the power weighted Hardy spaces

In this paper, we study the high‐dimensional Hausdorff type operators and establish their boundedness on the power weighted Hardy spaces for . As a consequence, we obtain that the Hausdorff type operator is bounded on if Φ is the Gauss function, or the Poisson function.     

Infinite products for power series

An algorithm is introduced and shown to lead to a unique infinite product representation for a given formal power series A(z) with A(0) = 1. The infinite product is , where all b_n ≠ 0, r_n , and r_n+1 > r_n. The degree of approximation by the polynomial (1 + b₁z^r₁) · · · (1 + b_nz^r_n) is also considered.     

Moment multipliers for power series

По определению после довательность {μ _n пр инадлежит классуG _s , если звезда М иттагЛеффлера произвольного степе нного ряда (1) $$\mathop \sum \limits_0^\infty a_n z^n , \mathop {lim sup}\limits_{n \to \infty } \left| {a_n } \right|^{1/n}< \infty $$ , совпадает со звёздам и Миттаг-Леффлера сте пенных рядов $$\mathop \sum \limits_0^\infty \mu _n a_n z^n ,\mathop \sum \limits_0^\infty \mu _n^{ - 1} a_n z^n $$ . В работе установлены следующие утвержден ия Теорема 1.Для произво льной последователь ности ? _n с условиями $$0< \varphi _n< 1,\mathop {lim}\limits_{n \to \infty } \varphi _n = 0,\mathop {lim}\limits_{n \to \infty } \varphi _n^{1/n} = 1$$ существует неубываю щая функция χ(t) такая, ч то моменты \(\mu _n = \int\limits_0^1 {t^n d\chi (t)} \) удовлетворяют условию 0<μ_nn звезда М иттаг-Леффлера любог о ряда (1) совпадает со звездой МиттагЛеффлера степенных рядов . Теорема 2. Для произвол ьной неотрицательно й последовательности {а_n} с условием {a _n } и для любой последов ательности {?_n} для к оторой 0n<1, \(\mathop {\lim }\limits_{n \to \infty } \varepsilon _n = 0\) сущест вуютπ={π _n }∈G _s и последовательнос ть {п_i} такие, что a_nμ_n≦1 (n≧n₀), \(a_{n_i } \mu _{\mu _i } \geqq exp( - \varepsilon _{n_i } )\) (i=1, 2, ...) и при эmom звезда Миттаг-Леффлера ряда (1) совпа дает со звездой Миттаг- Леффлера степенных р ядов .     

Investigation of Haar and Franklin series in Hardy spaces

(L ¹,H) (, ) , ; H — . , , L ¹ . [13] , . , , , .     

A new condition for certain trigonometric series

The present paper proposes a new natural weak condition to replace the quasimonotone and O-regularly varying quasimonotone conditions, and shows the conclusion of the main result in [10] still holds. Moreover, we prove that any O-regularly varying quasimonotone condition implies this condition, thus tne vague implication of quasimonotonicity can be avoided in practice.     

Boundedness of Hausdorff operators on the power weighted Hardy spaces

In this paper, we consider the two-dimensional Hausdorff operators on the power weighted Hardy space H_(|x|α)~1(R~2) ( -1 ≤α≤0), defined by H_(Φ,A)f(x)=∫R~2Φ(u)f(A(u)x)du,where Φ∈L_loc~1(R~2),A(u) = (α_(ij)(u))_(i,j=1)~2 is a 2×2 matrix, and each α_(i,j) is a measurablefunction.We obtain that HΦ,A is bounded from H_(|x|~α)~1(R~2) ( -1≤α≤0) to itself, if∫R2|Φ(u)‖det A~(-1)(u)|‖A(u)‖~(-α)ln(1+‖A~(-1)(u)‖~2/|det A~(-1)(u)|)du∞.This result improves some known theorems, and in some sense it is sharp.