Inequalities for absolute maxima of a polynomial and its derivative

Suppose z ₁, z ₂, ... z _n are complex numbers with absolute values more than 1 and Arg z _j Arg z _k for j k where Arg w stands for the argument of the complex number w in [0,2). In this note we show that

Approximation by Simplest Fractions

In this paper, a number of problems concerning the uniform approximation of complex-valued continuous functions on compact subsets of the complex plane by simplest fractions of the form are considered. In particular, it is shown that the best approximation of a function by the fractions is of the same order of vanishing as the best approximations by polynomials of degree .     

Duals of holomorphic Besov spaces on the polydisks and diagonal mappings

Let U ⁿ be the unit polydisk in C ⁿ and S be the space of functions of regular variation. Let 1 ≤ p < ∞, ω = (ω ₁, ..., ω _n ), ω _j ∈ S(1 ≤ j ≤ n) and f ∈ H(U ⁿ ). The function f is said to be in holomorphic Besov space B _p (ω) if

Second-Order Subelliptic Operators on Lie Groups I: Complex Uniformly Continuous Principal Coefficients

We consider second-order subelliptic operators with complex coefficients over a connected Lie group G. If the principal coefficients are right uniformly continuous then we prove that the operators generate strongly continuous holomorphic semigroups with kernels K satisfying Gaussian bounds. Moreover, the kernels are Hölder continuous and for each 0, 1 and > 0 one has estimates

An Elementary Approach to an Eigenvalue Estimate for Matrices

A celebrated result of Johnson, Maurey, König and Retherford from 1977 states that for every complex matrix satisfies the following eigenvalue estimate:

Proper holomorphic mappings between generalized pseudoellipsoids

Summary We characterize the existence of proper holomorphic mappings between pseudoellipsoids of the form . If f:()() is any such mapping, the existence of a subgroup of Aut (()) such that is shown equivalent to a condition on the branch locus of f.     

Bounded solutions for 119-01119-01119-01-problem in pseudo-Siegel domains

Summary We study the problem of the existence of bounded solutions for the equation on pseudo-Siegel domains when the data satisfies the condition .

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Sunto Si studia il problema dell'esistenza di soluzioni limitate per l'equazione sui domini pseudo-Siegel quando il dato soddisfa alla condizione .

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Work done within the Project 40% M.U.R.S.T. «Geometria Reale e Complessa».     

A Littlewood-Paley type inequality

In this note we prove the following theorem: Let u be a harmonic function in the unit ball and . Then there is a constant C = C(p, n) such that

Interlineation of functions of 2 variables on M (M≥2) lines with the highest algebraic precision

A general algorithm is proposed for constructing interlineation operators , x=(x₁, x₂) with the properties

Continuity Conditions for a Class of Gaussian Chaos Processes Related to Continuous Additive Functionals of Lévy Processes

Let be sequences of real numbers which are symmetric in k. Let be independent sequences of independent normal random variables with mean zero and variance one. For each fixed choice of we consider

Interpolation by Rational Functions with Nodes on the Unit Circle

From the Erds–Turán theorem, it is known that if f is a continuous function on and L _n (f, z) denotes the unique Laurent polynomial interpolating f at the (2 n + 1)th roots of unity, then Several years later, Walsh and Sharma produced similar result but taking into consideration a function analytic in and continuous on and making use of algebraic interpolating polynomials in the roots of unity.In this paper, the above results will be generalized in two directions. On the one hand, more general rational functions than polynomials or Laurent polynomials will be used as interpolants and, on the other hand, the interpolation points will be zeros of certain para-orthogonal functions with respect to a given measure on .     

On Invariance of Some Classes of Holomorphic Functions Under Integrodifferential Operators

The following classes of functions analytic in the unit disk are considered:

Boundedness and compactness of an integral operator in a mixed norm space on the polydisk

We study the following integral type operator

$T_g (f)(z) = \int\limits_0^{z_{} } { \cdots \int\limits_0^{z_n } {f(\zeta _1 , \ldots ,\zeta _n )} g(\zeta _1 , \ldots ,\zeta _n )d\zeta _1 , \ldots ,\zeta _n } $

in the space of analytic functions on the unit polydisk U ⁿ in the complex vector space ?ⁿ. We show that the operator is bounded in the mixed norm space

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, with p, q ∈ [1, ∞) and α = (α₁, …, α_n), such that α_j > ?1, for every j = 1, …, n, if and only if \(\sup _{z \in U^n } \prod\nolimits_{j = 1}^n {\left( {1 - \left| {z_j } \right|} \right)} \left| {g(z)} \right| < \infty \). Also, we prove that the operator is compact if and only if \(\lim _{z \to \partial U^n } \prod\nolimits_{j = 1}^n {\left( {1 - \left| {z_j } \right|} \right)} \left| {g(z)} \right| = 0\).

     

A Minimal-Automaton-Based Algorithm for the Reliability of Con(d,k, n) Systems

A d-within-consecutive-k-out-of-n system, abbreviated as Con(d, k, n), is a linear system of n components in a line which fails if and only if there exists a set of k consecutive components containing at least d failed ones. So far the fastest algorithm to compute the reliability of Con(d, k, n) is Hwang and Wright's algorithm published in 1997, where . In this paper we use automata theory to reduce to . For d small or close to k, we have reduced from exponentially many (in k) to polynomially many. The computational complexity of our final algorithm is , where .     

A Generalized Measure Concentrated on a Closed Set of Measure Zero

When E is a closed set of measure zero in the dyadic group and the Walsh series satisfies

Whitney Interpolation Constants Bounded by 2 for k = 5, 6, 7

Let f C[0, 1], k = 5, 6, 7. We prove that if f(i/(k – 1)) = 0, i = 0, 1,..., k – 1, then      

On a Binary Diophantine Inequality Involving Prime Numbers

Let 1 < c < 15/14 and N a sufficiently large real number. In this paper we prove that, for all (N, 2N\ A with , the inequality has solutions in primes .     

Sharp Constant in a Jackson-Type Inequality for Approximation by Positive Linear Operators

In what follows, C is the space of -periodic continuous real-valued functions with uniform norm, is the first continuity modulus of a function with step h, H _n is the set of trigonometric polynomials of order at most n, is the set of linear positive operators (i.e., of operators such that for every ), is the space of square-integrable functions on ,

On the LIL for Self-Normalized Sums of IID Random Variables

Let be i.i.d. random variables and let, for each and . It is shown that a.s. whenever the sequence of self-normalized sums S _n /V _n is stochastically bounded, and that this limsup is a.s. positive if, in addition, X is in the Feller class. It is also shown that, for X in the Feller class, the sequence of self-normalized sums is stochastically bounded if and only if