On the Norm of the Metric Projections

LetXbe a Banach space. GivenMa subspace ofXwe denote withP_Mthe metric projection ontoM. We defineπ(X) sup{P_M: Ma proximinal subspace ofX}. In this paper we give a bound forπ(X). In particular, whenX=L_p, we obtain the inequality P_M2^|2/p−1|, for every subspaceMofL_p. We also show thatπ(X)=π(X*).     

On accuracy of approximation of the spectral radius by the Gelfand formula

The famous Gelfand formula ρ(A)=limsup_n→∞A^n1/n for the spectral radius of a matrix is of great importance in various mathematical constructions. Unfortunately, the range of applicability of this formula is substantially restricted by a lack of estimates for the rate of convergence of the quantities A^n1/n to ρ(A). In the paper this deficiency is made up to some extent. By using the Bochi inequalities we establish explicit computable estimates for the rate of convergence of the quantities A^n1/n to ρ(A). The obtained estimates are then extended for evaluation of the joint spectral radius of matrix sets.     

Error bounds for the perturbation of the Drazin inverse under some geometrical conditions

This paper studies the Drazin inverse for perturbed matrices. For that, given a square matrix A, we consider and characterize the class of matrices B with index s such that , and , where and denote the null space and the range space of a matrix A, respectively, and A^D denote the Drazin inverse of A. Then, we provide explicit representations for B^D and BB^D, and upper bounds for the relative error B^D-A^D/A^D and the error BB^D-AA^D. A numerical example illustrates that the obtained bounds are better than others given in the literature.     

On -convergence of Fourier series

In this paper we consider the trigonometric Fourier series with the β-general monotone coefficients. Necessary and sufficient conditions of L₁-convergence for such a series, that is f−S_n=o(1), are obtained in terms of coefficients.     

On Mean Convergence of Lagrange Interpolation for General Arrays

For n1, let {x_jn}ⁿ_j=1 be n distinct points in a compact set K and letL_n[·] denote the corresponding Lagrange interpolation operator. Let v be a suitably restricted function on K. What conditions on the array {x_jn}_1jn, n1 ensure the existence of p>0 such that lim_n→∞ (f−L_n[f]) v_Lp(K)=0 for very continuous f: K→ ? We show that it is necessary and sufficient that there exists r>0 with sup_n1 π_nv_Lr(K) ∑ⁿ_j=1 (1/|π′_n| (x_jn))<∞. Here for n1, π_n is a polynomial of degree n having {x_jn}ⁿ_j=1 as zeros. The necessity of this condition is due to Ying Guang Shi.     

Weak Copositive and Intertwining Approximation

It is known that shape preserving approximation has lower rates than unconstrained approximation. This is especially true for copositive and intertwining approximations. ForfL_p, 1p<∞, the former only has rateω(f, n⁻¹)_p, and the latter cannot even be bounded byC f_p. In this paper, we discuss various ways to relax the restrictions in these approximations and conclude that the most sensible way is the so-calledalmostcopositive/intertwining approximation in which one relaxes the restriction on the approximants in a neighborhood of radiusΔ_n(y_j) of each sign changey_j.     

Strong characterizing sequences for subgroups of compact groups

In [A. Biró, V.T. Sós, Strong characterizing sequences in simultaneous Diophantine approximation, J. Number Theory 99 (2003) 405–414] we proved that if Γ is a subgroup of the torus R/Z generated by finitely many independent irrationals, then there is an infinite subset AZ which characterizes Γ in the sense that for γR/Z we have ∑_aAaγ<∞ if and only if γΓ. Here we consider a general compact metrizable Abelian group G instead of R/Z, and we characterize its finitely generated free subgroups Γ by subsets AG^*, where G^* is the Pontriagin dual of G. For this case we prove stronger forms of the analogue of the theorem of the above mentioned work, and we find necessary and sufficient conditions for a kind of strengthening of this statement to be true.     

An isomorphic characterization of L-spaces

We show that a sequentially (τ)-complete topological vector lattice X_τ is isomorphic to some L¹(μ), if and only if the positive cone can be written as X₊ = ₊B for some convex, (τ)-bounded, and (τ)-closed set B X₊ {0}. The same result holds under weaker hypotheses, namely the Riesz decomposition property for X (not assumed to be a vector lattice) and the monotonic σ-completeness (monotonic Cauchy sequences converge). The isometric part of the main result implies the well-known representation theorem of Kakutani for (AL)-spaces. As an application we show that on a normed space Y of infinite dimension, the “ball-generated” ordering induced by the cone Y₊ = ₊ (for u >) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials.     

Global existence, uniqueness, and continuous dependence for a semilinear initial value problem

In this paper, we consider the semilinear initial value problem associated with an operator A whose spectrum lies in a sector of the complex plane and whose resolvent satisfies (z−A)⁻¹M|z|^γ for some −1<γ<0 and all z outside the sector. The properties of existence and uniqueness of global mild solutions and continuous dependence on the initial data are investigated.