Minimality of root vectors of operator functions analytic in an angle

We study the minimality of elementsx _h,j,k of canonical systems of root vectors. These systems correspond to the characteristic numbers _k of operator functionsL() analytic in an angle; we assume that operators act in a Hilbert space . In particular, we consider the case whereL()=I+T()c, >0,I is an identity operator,C is a completely continuous operator, (I- C)^–1c for ¦arg¦, 0<<, the operator functionT() is analytic, and T()c for ¦arg¦<. It is proved that, in this case, there exists >0 such that the system of vectorsC ^v x _h,j,k is minimal in for arbitrary positive <1+, provided that ¦_k¦>.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 545–566, May, 1994.This research was partially supported by the Ukrainian State Committee of Science and Technology.     

Inversion and structure formulae for a generalized Laplace transform

, D_v(.) , .     

On power series with positive coefficients

M. . , . , ^p () (). , , .     

The survival of the large dimensional basic contact process

Summary We consider the d-dimensional basic contact process obtaining the limit value of the probability of survival when d+, and showing that the finite dimensional distributions of the upper invariant measure become of the product form as d+.     

Holomorphic Solutions of Some Functional Equations II

We study a certain type of functional equation, which is of significance from the view point of systems of difference equations. Let the characteristic values of the system be and The case that either || > 1 or 0 < || < 1 has been treated in a former paper. The case that = 1, || = 1 with 1 will be given in another paper. The present note deals with the case = = 1, the most difficult case.AMS Subject Classification (1991): 39A10 39B05     

On the almost everywhere convergence of double Fourier series of integrable functions

, . . Q _k [0,2],k=1,2, — . F(x, y)L(T), T=[0, 2]², G(x, y)L(T) , G(x,y)=F(x,y) Q=Q ₁ ×Q ₂ - .     

Converse theorems for multidimensional Kantorovich operators

L _p [0, l]. . . - .

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Supported by National Science Foundation, Zhejiang Provincial Science Foundation of China, and Alexander von Humboldt Foundation of Germany.     

Morera type theorems on the unit disc

, . . : , , , , .     

Pointwise convergence of expansions with respect to certain product systems

- , - _n A_n-,. , . , ¦_n¦=1 (n=1,2, ), . , — — , .     

High order approximation of implicitly defined maps

Summary Approximations for the function implicitly defined by (u)=(u, (u)) are obtained via the iterative scheme _n(u)=(u, _n–1(u)). In this paper the uniform convergence of high order derivatives of _n to the corresponding derivatives of is proved. This result yields a high order approximation theorem for the input-output map generated by a nonlinear control system, using linear combinations of iterated integrals of the control.Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

Eine Familie interpolatorischer Quadraturformeln mit ableitungsfreien Fehlerschranken

Zusammenfassung Es wird eine von einem Parameter , 01 abhängige Familie von Quadraturformeln vorgestellt, für die auf gewissen Hilberträumen analytischer Funktionen ableitungsfreie Fehlerschdranken existieren. Für =0 erhält man als Spezialfall die Gaußschen Quadraturformeln und für =1 die Wilfschen Quadraturformeln. Sämtliche Formeln haben folgende Eigenschaften: Sie sind interpolatorisch, d.h. hier, sie können durch Integration eines hermiteschen Interpolationsoperators erzeugt werden, sie haben positive Gewichte. Ihre Stützstellen liegen im Innern des Integrationsintervalles. Sie sind auch erzeugbar durch Minimierung des Fehlerkoeffizienten, und sie konvergieren für jede im Integrationsintervall stetige Funktion.

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A family of quadrature formulas having error bounds without derivatives

Summary A family of quadrature formulas depending on a parameter , 01 is presented which admits error estimates without derivatives for certain Hilbert spaces of analytic functions. For =0 the Gaussian quadrature formulas are contained as a special case, and for =1 the same is valid for the Wilfian quadrature formulas. The formulas have the following properties: They are interpolatory formulas, that means they may be generated on integration of a Hermitian interpolating operator, they have positive weights. Their nodes are inside the interval of integration, and they also may be produced by minimizing the error coefficient. They are convergent for every function which is continuous between the limits of integration.

     

Exceptional sets and wavelet packets orthonormal bases

We give a partial positive answer to a problem posed by Coifman et al. in [1]. Indeed, starting from the transfer function m₀ arising from the Meyer wavelet and assuming m₀=1 only on [–/3, /3], we provide an example of pairwise disjoint dyadic intervals of the form I(n, q)=[2^qn, 2^q(n+1)), (n, q)EN×Z, which cover [0, +) except for a set A of Hausdorff dimension equal to 1/2, and such that the corresponding wavelet packets 2^q/2w_n (2^qx–k), kZ, (n, q)EN×Z form an orthonormal basis of L²(R).     

New improved error bounds for the linear complementarity problem

New local and global error bounds are given for both nonmonotone and monotone linear complementarity problems. Comparisons of various residuals used in these error bounds are given. A possible candidate for a best error bound emerges from our comparisons as the sum of two natural residuals.This material is based on research supported by Air Force Office of Scientific Research Grant AFOSR-89-0410 and National Science Foundation Grant CCR-9101801.     

A note on a set-mapping problem of Hajnal and Máté

It is consistent that there exists a set mappingF with <F(, )< for + 2 >w ₂ with no uncountable free sets.Research supported by Hungarian National Research Fund No. 1805 and 1908.     

Blow-up of solutions of nonlinear wave equations in three space dimensions

Let u(x,t) be a solution, uA|u|^p for xIR₃, t0 where is the d'Alembertian, and A, p are constants with A>0, 10–|x–x⁰|, if the initial data u(x,0), u_t(x,0) have their support in the ball |x–x⁰|t₀. In particular global solutions of u=A|u|^p with initial data of compact support vanish identically. On the other hand for A>0, p>1+2 global solutions of u=A|u|^p exist, if the initial data are of compact support and u is sufficiently small in a suitable norm. For p=2 the time at which u becomes infinite is of order u^–2.Dedicated to Hans Lewy and Charles B. Morrey, Jr.The research for this paper was performed at the Courant Institute and supported by the Office of Naval Research under Contract No. N00014-76-C-0301. Reproduction in whole or part is permitted for any purpose of the United States Government.     

Minimal coverings and maximal packings of (k−1)-subsets by k-subsets

This paper studies the asymptotic behavior of functions M(n, k, k–1, ) and m(n, k, k–1, ), equal to the respective cardinalities of the minimal -covering and maximal -packing of all (k–1)-subsets of the n-element set of its k-subsets. It is shown that, if sequence k=k(n) is such that k(n)/n 0 as then m(n, k, k–1, ).( _k–1 ⁿ ).k¹, and if as n , thenM(n,k,k–1,).( _k–1 ⁿ ).k^–1. A consequence of these results is the validity of the Erdös-Hanani conjecture concerning the asymptotic behavior of functions M(n, k, k–1, 1) and m(n, k, k–1, 1).Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 565–571, April, 1977.In conclusion, the author wishes to thank A. A. Sapozhenko under whose direction the present work was achieved.     

On aU-set for multiple Walsh series

m- (m2) , - , .     

L 2 error estimates for finite elements with interpolated boundary conditions

Summary TheL ² error in the nearly zero quadratic approximate solution to Poisson's equation is shown to be of optimal order. The same method of proof can be used to show that theL ² error with a space of cubics developed by Ridgway Scott is optimal.This work was supported by the Office of Naval Research under Contract Number ONR NR-044-453 and by the Naval Ordnance Laboratory Independent Research Fund.