Into isomorphisms in tensor products of Banach spaces

We establish quantitative extensions of two Grothendieck's results on into isomorphisms in projective tensor products. Among others, we prove the following. Let Y be a closed subspace of a Banach space Z and let j : Y → Z denote the identity embedding. If Y is complemented in its bidual Y??, then the injection modulus of the natural inclusion Id ? j : Y??Y → Y??Z satisfies 1/λ _loc (Y,Z) ≤ i(Id ? j) ≤ λ(Y,Y??)/λ(Y,Z), where λ(·,·) and λ_loc(·,·) are, respectively, the projection and the local projection constants.     

Some results on nearest points in normed linear spaces

We will focus on some results that we hope to give an algorithm for constructing the best approximations in some types of normed linear spaces. Also some results on best approximation will be obtained.     

Direct local and global approximation theorems for some linear positive operators

In this paper we establish direct local and global approximation theorems for Baskakov type operators and Szasz - Mirakjan type operators, respectively.     

BERGMAN TYPE PROJECTIONS ON L^p SPACES

The paper is a survey on the action of Bergman type projections on various Lp on three types of holomorphic function spaces： weighted Bergman spaces, the Bloch spaces. The focus is space, and diagonal Besov spaces.     

COMPUTING KARMARKAR＇S PROJECTIONS IN STOCHASTIC LINEAR PROGRAMMING

The Karmarkar‘s projections are computed quickly by using Moore-Penrose g inverse and matrix factorization in stochastic linear programming. So computation work of (AD^2A^T)^-1 is decreased.     

Local linear convergence of approximate projections onto regularized sets

The numerical properties of algorithms for finding the intersection of sets depend to some extent on the regularity of the sets, but even more importantly on the regularity of the intersection. The alternating projection algorithm of von Neumann has been shown to converge locally at a linear rate dependent on the regularity modulus of the intersection. In many applications, however, the sets in question come from inexact measurements that are matched to idealized models. It is unlikely that any such problems in applications will enjoy metrically regular intersection, let alone set intersection. We explore a regularization strategy that generates an intersection with the desired regularity properties. The regularization, however, can lead to a significant increase in computational complexity. In a further refinement, we investigate and prove linear convergence of an approximate alternating projection algorithm. The analysis provides a regularization strategy that fits naturally with many ill-posed inverse problems, and a mathematically sound stopping criterion for extrapolated, approximate algorithms. The theory is demonstrated on the phase retrieval problem with experimental data. The conventional early termination applied in practice to unregularized, consistent problems in diffraction imaging can be justified fully in the framework of this analysis providing, for the first time, proof of convergence of alternating approximate projections for finite dimensional, consistent phase retrieval problems.     

Incomplete oblique projections for solving large inconsistent linear systems

The authors introduced in previously published papers acceleration schemes for Projected Aggregation Methods (PAM), aiming at solving consistent linear systems of equalities and inequalities. They have used the basic idea of forcing each iterate to belong to the aggregate hyperplane generated in the previous iteration. That scheme has been applied to a variety of projection algorithms for solving systems of linear equalities or inequalities, proving that the acceleration technique can be successfully used for consistent problems. The aim of this paper is to extend the applicability of those schemes to the inconsistent case, employing incomplete projections onto the set of solutions of the augmented system Ax − r = b. These extended algorithms converge to the least squares solution. For that purpose, oblique projections are used and, in particular, variable oblique incomplete projections are introduced. They are defined by means of matrices that penalize the norm of the residuals very strongly in the first iterations, decreasing their influence with the iteration counter in order to fulfill the convergence conditions. The theoretical properties of the new algorithms are analyzed, and numerical experiences are presented comparing their performance with several well-known projection methods. Dedicated to Clovis Gonzaga on the occassion of his 60th birthday.     

Coincidences of projections and linear n-valued maps of tori

We prove that the Nielsen fixed point number N(φ) of an n-valued map φ:X?X of a compact connected triangulated orientable q-manifold without boundary is equal to the Nielsen coincidence number of the projections of the graph of φ, a subset of X×X, to the two factors. For certain q×q integer matrices A, there exist “linear” n-valued maps Φn,A,σ:Tq?Tq of q-tori that generalize the single-valued maps fA:Tq→Tq induced by the linear transformations TA:Rq→Rq defined by TA(v)=Av. By calculating the Nielsen coincidence number of the projections of its graph, we calculate N(Φn,A,σ) for a large class of linear n-valued maps.     

On the spectrum of linear combinations of two projections in C*-algebras

In this note, we study the spectrum and give estimations for the spectral radius of linear combinations of two projections in C*-algebras. We also study the commutator of two projections.     

Densities for unbalanced measures of a linear Cantor set

Define a linear Cantor set C to be the attractor of a linear iterated function system fj （x） =rjx ＋ bj（j = 1,2,…,N）, on the line satisfying the sures with respect to C,we study the centered upper and the centered lower density for Ф（t） = t^s withunnatural choices and with natural choices of s.     

Boundedness of multilinear operators on weighted herz type spaces

In this paper, the author establishes the boundedness of multilinear operators on weighted Herz spaces and Herz-type Hardy spaces. The author also obtains their weak estimates on endpoints. As a special case, the conclusions may lead to the weighted estimates for multilinear Calderon-Zygmund operators.     

Pole placement controllers for linear time-delay systems with commensurate point delays

This paper investigates the exact and approximate spectrum assignment properties associated with realizable output-feedback pole-placement type controllers for single-input single-output linear time-invariant time-delay systems with commensurate point delays. The controller synthesis problem is discussed through the solvability of a set of coupled diophantine equations of polynomials. An extra complexity is incorporated to the above design to cancel extra unsuitable dynamics being generated when solving the above diophantine equations. Thus, the complete controller tracks any arbitrary prefixed (either finite or delaydependent) closed-loop spectrum. However, if the controller is simplified by deleting the above mentioned extra complexity, then the robust stability and approximated spectrum assignment are still achievable for a certain sufficiently small amount of delayed dynamics. Finally, the approximate spectrum assignment and robust stability problems are revisited under plant disturbances if the nominal controller is maintained. In the current approach, the finite spectrum assignment is only considered as a particular case to the designer‘s choice of a (delay-dependent) arbitrary spectrum assignment objective.     

L P boundedness of area function associated with operators on product spaces

Let L be the infinitesimal generator of an analytic semigroup on L2 (Rn) with suitable upper bounds on its heat kernels. Assume that L has a bounded holomorphic functional calculus on L2(Rn). In this paper,we define the Littlewood- Paley g function associated with L on Rn × Rn, denoted by GL(f)(x1, x2), and decomposition, we prove that ‖SL(f)‖p ≈‖GL(f)‖p ≈‖f‖p for 1 ＜ p ＜∞.     

Boundedness of higher order commutators of generalized fractional integral operators on Hardy spaces

Let Tμ,b,m be the higher order commutator generated by a generalized fractional integral operator Tμ and a BMO function b. In this paper, we will study the boundedness of Tμ,b,m on classical Hardy spaces and Herz-type Hardy spaces.