On Minimizing and Stationary Sequences of a New Class of Merit Functions for Nonlinear Complementarity Problems

Motivated by the work of Fukushima and Pang (Ref. 1), we study the equivalent relationship between minimizing and stationary sequences of a new class of merit functions for nonlinear complementarity problems (NCP). These merit functions generalize that obtained via the squared Fischer–Burmeister NCP function, which was used in Ref. 1. We show that a stationary sequence {x^k} /Reⁿ is a minimizing sequence under the condition that the function value sequence {F(x ^k)} is bounded above or the Jacobian matrix sequence {F(x ^k)} is bounded, where F is the function involved in NCP. The latter condition is also assumed by Fukushima and Pang. The converse is true under the assumption of {F(x ^k)} bounded. As an example shows, even for a bounded function F, the boundedness of the sequence {F(x ^k)} is necessary for a minimizing sequence to be a stationary sequence.     

Hankel operators on the Bergman spaces of strongly pseudoconvex domains

We characterize functions fL ²(D) such that the Hankel operators H_f are, respectively, bounded and compact on the Bergman spaces of bounded strongly pseudoconvex domains.Research partially supported by a grant of the National Science Foundation.     

Characterizations of simultaneous farthest point in normed linear spaces with applications

In this paper, we consider a problem of best approximation (simultaneous farthest point) for bounded sets in a real normed linear space X. We study simultaneous farthest point in X by elements of bounded sets, and present various characterizations of simultaneous farthest point of elements by bounded sets in terms of the extremal points of the closed unit ball of X ^*, where X ^* is the dual space of X. We establish the characterizations of simultaneous farthest points for bounded sets in , the space of all real-valued continuous functions on a compact topological space Q endowed with the usual operations and with the norm . It is important to state clearly that the contribution of this paper in relation with the previous works (see, for example, [9, Theorem 1.13]) is a technical method to represent the distance from a bounded set to a compact convex set in X which specifically concentrates on the Hahn-Banach Theorem in X.     

Traces of bounded analytic functions on finite unions of Carleson sets

The traces of functions from H on the union HT of n Carleson subsets of the unit circle form a space of functions on HT for which the first n — 1 divided differences (with respect to the hyperbolic metric) are uniformly bounded.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 31–34, 1983.     

Liouville-type theorem for the drifting Laplacian operator

Given manifolds with a smooth measure (M, g, e ^?f dV), we consider gradient estimates for positive harmonic functions of the drifting Laplacian. If the ∞-Bakry-Emery Ricci tensor is bounded from below and \({|\nabla f|}\) is bounded, we obtain a Liouville-type theorem. This extends a classical result of Cheng and Yau.     

Approximation of differentiable functions by partial sums of their Fourier series

For the classes of differentiable functions W ^r , r > 0, which include the classes of functions which have derivativesf(r) or (r)with moduli bounded by one, we obtain an asymptotic formula for the supremum of the difference between a function and the partial sums of its Fourier series. The remainder term in our formula is Cn^–r, in which C is a constant.Translated from Matematicheskii Zametki, Vol. 4, No. 3, pp. 291–300, September, 1068.     

Regularized spectral distributions

For C a bounded, injective operator with dense image, we define a C-regularized spectral distribution. This produces a functional calculus, f f(B), from C() into the space of closed densely defined operators, such that f(B)C is bounded when f has compact support. As an analogue of Stone's theorem, we characterize certain regularized spectral distributions as corresponding to generators of polynomially bounded C-regularized groups. We represent the regularized spectral distribution in terms of the regularized group and in terms of the C-resolvent. Applications include the Schrödinger equation with potential, and symmetric hyperbolic systems, all on L^p(ⁿ) (1p<), C _o(ⁿ), BUC(ⁿ), or any space of functions where translation is a bounded strongly continuous group.     

Characterization of finite unions of Carleson sets in terms of solvability of interpolational problems

The following assertion is proved: a discrete subset of the unit disk can be represented as the union of n Carleson sets if and only if the traces of functions from H on form a space of functions on for which the first n–1 divided differences (with respect to the hyperbolic metric) are uniformly bounded.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 31–35, 1984.     

A note on sharply bounded arithmetic

Summary We prove some independence results for the bounded arithmetic theoryR ₂ ⁰ , and we define a class of functions that is shown to be an upper bound for the class of functions definable by a certain restricted class of ₁ ^b in extensions ofR ₂ ⁰ .Mathematics subject classification: 03F30This article was processed by the author using the LATEX style filepljour1 from Spinger-Verlag.     

Cardinal invariants concerning bounded families of extendable and almost continuous functions

In this paper we introduce and examine a cardinal invariant closely connected to the addition of bounded functions from to . It is analogous to the invariant defined earlier for arbitrary functions by T. Natkaniec. In particular, it is proved that each bounded function can be written as the sum of two bounded almost continuous functions, and an example is given that there is a bounded function which cannot be expressed as the sum of two bounded extendable functions.

     

Continuity of the metric-projection operators

We consider the metric-projection operators in the space LP(), 1 < p < , where M^b, M^b being the space of bounded Randon measures. The metric-projection operators project a fixed element g C^b in LP() onto a closed convex subset K C_u ^b, where C_u ^b is the space of bounded continuous functions in the topology of uniform convergence. We obtain statements on the continuity of the metric-projection operators, considered as mappings from M^b into K.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 8, pp. 1141–1144, August, 1992.     

Cores of Second Order Differential Linear Operators with Unbounded Coefficients on ℝN

We consider the problem of the existence of bi-cores for some classes of second order elliptic differential operators with unbounded coefficients generating bi-continuous semigroups on the space of bounded continuous functions on ^N.     

Injectivity sets for spherical means on the Heisenberg group

We prove that the boundary of a bounded domain is a set of injectivity for the twisted spherical means on ⁿ for a certain class of functions on ⁿ . As a consequence we obtain results about injectivity of the spherical mean operator in the Heisenberg group and the complex Radon transform.     

Supports of fourier transforms of refinable frame functions and their applications to FMRA

Let M be a d × d expansive matrix, and FL ²(??) be a reducing subspace of L ²(? ^d ). This paper characterizes bounded measurable sets in ? ^d which are the supports of Fourier transforms of M-refinable frame functions. As applications, we derive the characterization of bounded measurable sets as the supports of Fourier transforms of FMRA (W-type FMRA) frame scaling functions and MRA (W-type MRA) scaling functions for FL ²(??), respectively. Some examples are also provided.     

Convoluted C-cosine functions and semigroups. Relations with ultradistribution and hyperfunction sines

Convoluted C-cosine functions and semigroups in a Banach space setting extending the classes of fractionally integrated C-cosine functions and semigroups are systematically analyzed. Structural properties of such operator families are obtained. Relations between convoluted C-cosine functions and analytic convoluted C-semigroups, introduced and investigated in this paper are given through the convoluted version of the abstract Weierstrass formula which is also proved in the paper. Ultradistribution and hyperfunction sines are connected with analytic convoluted semigroups and ultradistribution semigroups. Several examples of operators generating convoluted cosine functions, (analytic) convoluted semigroups as well as hyperfunction and ultradistribution sines illustrate the abstract approach of the authors. As an application, it is proved that the polyharmonic operator Δn², n∈N, acting on L²[0,π] with appropriate boundary conditions, generates an exponentially bounded Kn-convoluted cosine function, and consequently, an exponentially bounded analytic Kn+1-convoluted semigroup of angle , for suitable exponentially bounded kernels Kn and Kn+1.     

Sharp determinants

We introduce a sharp trace Tr^# and a sharp determinant Det^#(1-z) for an algebra of operators acting on functions of bounded variation on the real line. We show that the zeroes of the sharp determinant describe the discrete spectrum of . The relationship with weighted zeta functions of interval maps and Milnor-Thurston kneading determinants is explained. This yields a result on convergence of the discrete spectrum of approximated operators.Oblatum 8-V-1995 & IX-1995On leave from CNRS, UMR 128, ENS Lyon, France     

Singular integral equations — The convergence of the Nyström interpolant of the Gauss-Chebyshev method

Nyström's interpolation formula is applied to the numerical solution of singular integral equations. For the Gauss-Chebyshev method, it is shown that this approximation converges uniformly, provided that the kernel and the input functions possess a continuous derivative. Moreover, the error of the Nyström interpolant is bounded from above by the Gaussian quadrature errors and thus convergence is fast, especially for smooth functions. ForC input functions, a sharp upper bound for the error is obtained. Finally numerical examples are considered. It is found that the actual computational error agrees well with the theoretical derived bounds.This research has been partially supported by a grant from the Rutgers Research Council.     

A Besov class functional calculus for bounded holomorphic semigroups

It is well-known that -sectorial operators generally do not admit a bounded H^∞ calculus over the right half-plane. In contrast to this, we prove that the H^∞ calculus is bounded over any class of functions whose Fourier spectrum is contained in some interval [ε,σ] with 0<ε<σ<∞. The constant bounding this calculus grows as as and this growth is sharp over all Banach space operators of the class under consideration. It follows from these estimates that -sectorial operators admit a bounded calculus over the Besov algebra of the right half-plane. We also discuss the link between -sectorial operators and bounded Tadmor-Ritt operators.     

Representation of measurable positive definite generalized Toeplitz Kernels inR

We define the signature of a bounded operatorA onL ² S ² and prove thatA is smooth for the action ofSO(3) onL ² S ² if and only if its signature is smooth and any finite application of certain differential operators to it yields the signature of a bounded operator. Moreover, we show that the formal Fourier multipliers with bounded and smoothly variable coefficients are well defined bounded operators which areSO(3)-smooth.