Normal projection: deterministic and probabilistic algorithms

We consider the following problem： for a collection of points in an n-dimensional space, find a linear projection mapping the points to the ground field such that different points are mapped to different values. Such projections are called normal and are useful for making algebraic varieties into normal positions. The points may be given explicitly or implicitly and the coefficients of the projection come from a subset S of the ground field. If the subset S is small, this problem may be hard. This paper deals with relatively large S, a deterministic algorithm is given when the points are given explicitly, and a lower bound for success probability is given for a probabilistic algorithm from in the literature.     

Integral operators on analytic Morrey spaces

In this note, we characterize the boundedness of the Volterra type operator T _g and its related integral operator I _g on analytic Morrey spaces. Furthermore, the norm and essential norm of those operators are given. As a corollary, we get the compactness of those operators.     

上半平面上Bergman投影的有界性

利用再生核定义了Bergman投影算子,给出了Bergman投影算子有界性的充分条件.     

On polars of mixed projection bodies

Recently, Lutwak established general Minkowski inequality, Brunn-Minkowski inequality and Aleksandrov-Fenchel inequality for mixed projection bodies. In this paper, following Lutwak, we established their polar forms. As applications, we prove some interrelated results.     

Infinite systems of linear equations for real analytic functions

We study the problem when an infinite system of linear functional equations

has a real analytic solution on for every right-hand side and give a complete characterization of such sequences of analytic functionals . We also show that every open set has a complex neighbourhood such that the positive answer is equivalent to the positive answer for the analogous question with solutions holomorphic on .

     

Mixed projection methods for systems of variational inequalities

Let H be a real Hilbert space. Let be bounded and continuous mappings where D(F) and D(K) are closed convex subsets of H. We introduce and consider the following system of variational inequalities: find [u ^*,v ^*]∈D(F) × D(K) such that This system of variational inequalities is closely related to a pseudomonotone variational inequality. The well-known projection method is extended to develop a mixed projection method for solving this system of variational inequalities. No invertibility assumption is imposed on F and K. The operators K and F also need not be defined on compact subsets of H.      

The commutant of analytic Toeplitz operators on Bergman space

In this paper, using the matrix skills and operator theory techniques we characterize the commutant of analytic Toeplitz operators on Bergman space. For f（z） = z^ng（z） （n ≥1）, g（z） = b0 ＋ b1z^p1 ＋b2z^p2 ＋.. , bk ≠ 0 （k = 0, 1, 2,...）, our main result is =A′（Mf） = A′（Mzn）∩A′（Mg） = A′（Mz^s）, where s = g.c.d.（n,p1,p2,...）. In the last section, we study the relation between strongly irreducible curve and the winding number W（f,f（α））, α ∈ D.     

Uniform approximations of Stieltjes functions by orthogonal projection on the set of rational functions

Let μ be a positive Borel measure having support supp μ ⊂ [1, ∞) and satisfying the conditionf(t−1)⁻¹dμ(t)<∞. In this paper we study the order of the uniform approximation of the function

Conditional expectation and the Bergman projection

For any σ-algebra of measurable subsets of the unit disk generated by a finite Blaschke product, we prove that the associated conditional expectation operator commutes with the Bergman projection operator if and only if the σ-algebra is generated by a monomial. In the process, a formula for the conditional expectation operator (under certain assumptions) is obtained. When compared with earlier results of A.B. Aleksandrov concerning conditional expectation associated with σ-algebras of measurable subsets of the circle, our results exhibit a stark contrast between the way conditional expectation operators act in the Bergman and Hardy space settings.     

Projective analytic vectors and infinitesimal generators

We establish the notion of a “projective analytic vector”, whose defining requirements are weaker than the usual ones of an analytic vector, and use it to prove generation theorems for one-parameter groups on locally convex spaces. More specifically, we give a characterization of the generators of strongly continuous one-parameter groups which arise as the result of a projective limit procedure, in which the existence of a dense set of projective analytic vectors plays a central role. An application to strongly continuous Lie group representations on Banach spaces is given, with a focused analysis on concrete algebras of functions and of pseudodifferential operators.     

The structure of instantaneous frequencies of periodic analytic signals

In this paper, the structure of analytic signals is investigated by means of the relation between analytic signals and functions in the Hardy space. It is shown that an analytic signal is made up of two parts, one depending on the amplitude of the signal and another on the boundary value of an inner function. Based on this result, properties of the instantaneous frequencies of these two parts are studied, and it is found that negative instantaneous frequencies are caused by the amplitude of a signal. Finall...     

Directionally nondifferentiable metric projection

A closed convex set inR ² is constructed such that the associated metric projection onto that set is not everywhere directionally differentiable.     

On the convergence of projection methods: Application to the decomposition of affine variational inequalities

In this paper, we first discuss the global convergence of symmetric projection methods for solving nonlinear monotone variational inequalities under a cocoercivity assumption. A similar analysis is applied to asymmetric projection methods, when the mapping is affine and monotone. Under a suitable choice of the projection matrix, decomposition can be achieved. It is proved that this scheme achieves a linear convergence rate, thus enhancing results previously obtained by Tseng (Ref. 1) and by Luo and Tseng (Ref. 2).The research of the first author was supported by NSERC Grant A5789 and DND-FUHBP. The research of the second author was supported by NSERC Grant OGP-0157735.The authors are indebted to the referees and Associate Editor P. Tseng for their constructive comments.     

Inequalities for polars of mixed projection bodies

In 1993 Lutwak established some analogs of the Brunn-Minkowsi inequality and the Aleksandrov-Fenchel inequality for mixed projection bodies. In this paper, following Lutwak, we give their polars forms. Further, as applications of our methods, we give a generalization of Pythagorean inequality for mixed volumes.     

A successive projection method

It is of both theoretical and practical interest to find the projection of a point to the intersection of a finite number of closed convex sets by a sequence of projections to the individual sets successively. In this paper we study such a method and analyze its convergence properties. A main feature of the method is its capability to decompose the projection problem into several small ones. For some structured sparse problems these small subproblems can be solved independently and the presented method has a potential use in parallel computation.This material is based on work supported in part by the National Science Foundation under Grant DMS-8602419 and by the Center for Supercomputing Research and Development, University of Illinois.     

Parameter dependence of solutions of partial differential equations in spaces of real analytic functions

Let be an open set and let denote the class of real analytic functions on . It is proved that for every surjective linear partial differential operator and every family depending holomorphically on there is a solution family depending on in the same way such that The result is a consequence of a characterization of Fréchet spaces such that the class of ``weakly' real analytic -valued functions coincides with the analogous class defined via Taylor series. An example shows that the analogous assertions need not be valid if is replaced by another set.

     

The divisor of a generalized analytic function

For generalized analytic functions, we define the order of a zero and prove the following division theorem: if the orders of zeros of the numerator majorize those of the denominator, then the fraction is a generalized analytic function. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 655–661, May, 1997. Translated by V. E. Nazaikinskii