On Interpolation of the Fourier Maximal Operator in Orlicz Spaces

Let and be positive increasing convex functions defined on [0, ). Suppose satisfies the 2-condition, that is, (t)2 (C1t) for sufficiently large t, and has some nice properties. If -1(u)log(u+1) C2-1(u) for sufficiently large uthen we have S*(f) L CfL for all f L ([-, ])where S*(f) is the majorant function of partial sums of trigonometric Fourier series and fL is the Orlicz norm of f. This result is sharp.     

Continuous convolution semigroups integrating a submultiplicative function

Let be a submultiplicative function on a locally compact group G and let S be a convolution semigroup on G with Lévy measure . It is shown that the measures of S integrate if and only if integrates outside some neighbourhood of the identity of G (Theorem 1). Moreover if (X(t);t0) is the G-valued process with independent increments associated with the semigroup S it is shown that the measures of S integrate if and only if the random variable sup { (X(t)):0t1} is integrable (Theorem 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.     

Einschließungsaussagen für gewöhnliche Differentialoperatoren

Summary For differential operatorsM of second order (as defined in (1.1)) we describe a method to prove Range-Domain implications—Mu–u and an algorithm to construct these functions , , , . This method has been especially developed for application to non-inverse-positive differential operators. For example, for non-negativea ₂ and for given functions = we require =C ₀[0, 1] C ₂([0, 1]–T) whereT is some finite set), (M) (t)(t), (t[0, 1]–T) and certain additional conditions for eachtT. Such Range-Domain implications can be used to obtain a numerical error estimation for the solution of a boundary value problemMu=r; further, we use them to guarantee the existence of a solution of nonlinear boundary value problems between the bounds- and .     

О полноте систем функ ций и систем ортогона льных многочленов

The paper is devoted to the study of completeness problem of systems { ⁿ (x)} _n=0 inL ^p (a, b), where –a<b+,(x) is a weight function subject to mild assumptions, and(x) is a continuous function on (a,b), either bounded or unbounded in the neighbourhood of the end-points of (a,b). It turns out that this problem is connected with that of quasianalyticity of certain additive set of functions at a given point. As the most important application of the general results, the completeness problem is treated for systems of orthogonal polynomials.     

Stabilization of ill-posed Cauchy problems for parabolic equations

Summary In this paper we study the noncharacteristic Cauchy problem, u_t–(a(x)u_x)_x=0, x(0, l), t.(0, T], u(0, t)=(t), u_x(0,t)=0, 0tT, assuming only L for a. In the case of weak a priori bounds on u, we derive stability estimates on u of Hölder type in the interior and of logarithmic type at the boundary. Also the continuous dependence on a is considered.

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Sunto Nel presente lavoro consideriamo il problema di Cauchy non ben posto u_t= (a(x)u_x)_x, x(0, l), t(0, T), u(0, t)=(t), u_x(0, t)=0, 0tT. Supponiamo che a sia misurabile e limitato inferiormente e superiormente da constanti positive. Introduciamo delle limitazioni a priori su u e dimostriamo la dipendenza continua di u rispetto al dato sia in (0, l)×(0, T) (di tipo hölderiano) sia per x=l (di tipo logaritmico). Consideriamo, inoltre, la dipendenza continua di u da a.

     

Wave operators and similarity of certain non-normal one-parameter groups

This paper starts from a self-adjoint Schrödinger operator H(0) for three particles. If the interaction is dilation-analytic, H(0) has an analytic continuation H() (>0). G(t,) (–(±,a,) defined as strong limits, when t±, of t-dependent operators. The wave operators establish transformations under which the subgroups are similar to unitary groups. The scattering matrix determined by G(t,) is diagonal with respect to a.This work was supported in part by the National Science Foundation under grant DMS-8301096.     

The arithmetic of the characteristic Pólya functions

In the framework of the theory of D. Kendall's delphic semigroups are considered problems of divisibility in the semigroup of convex characteristic functions on the semiaxis (0,). Letn ()={:₁¦₁1 or ₁=}, and I_o()={: ₁¦ ₁ N()}. The following results are proved: 1) The semigroup is almost delphic in the sense of R. Davidson. 2) N() is a set of the type G which is dense in (in the topology of uniform convergence on compacta). 3) The class I_o() contains only the function identically equal to one.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 717–725, May, 1977.The author thanks I. V. Ostrovskii for the formulation of the problem and valuable remarks.     

Nonnegative Hall Polynomials

The number of subgroups of type and cotype in a finite abelian p-group of type is a polynomialg with integral coefficients. We prove g has nonnegative coefficients for all partitions and if and only if no two parts of differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certain order-preserving surjections that associate to each subgroup a vector dominated componentwise by . The nonzero components of (H) are the parts of , the type of H; if no two parts of differ by more than one, the nonzero components of – (H) are the parts of , the cotype of H. In fact, we provide an order-theoretic characterization of those isomorphism types of finite abelian p-groups all of whose Hall polynomials have nonnegative coefficients.     

A lower closure theorem for abstract control problems withL p-bounded controls

A lower closure theorem for an abstract control problem is proved. The functional isJ(,u)= _G f ⁰(t, (M)(t),u(t))dt and the state equations areN(t)=f(t, (M)(t),u(t)). It is shown that, if {( _k ,u _k)} is a sequence of admissible controlsu _k and corre-sponding trajectories _k such that lim infJ( _k ,u _k)<+ and such that _k weakly,M _k M strongly,N _k N weakly, and {u _k} is bounded in someL _p norm, then there is a controlu such that (,u) is admissible and lim infJ( _k ,u _k)J(,u).Dedicated to Professor M. R. HestenesThis research was supported by the National Science Foundation, Grant No. GP-33551X.     

Über eine Verallgemeinerung der Nicoletti-Aufgabe für Funktional-Differentialgleichungen mit voreilendem Argument

We consider a functional differential equation (1) (t)=F(t,) fort[0,+) together with a generalized Nicoletti condition (2)H()=. The functionF: [0,+)×C ⁰[0,+)B is given (whereB denotes the Banach space) and the value ofF (t, ) may depend on the values of (t) fort[0,+);H: C ⁰[0,+)B is a given linear operator and B. Under suitable assumptions we show that when the solution :[0,+)B satisfies a certain growth condition, then there exists exactly one solution of the problem (1), (2).     

Isometries of symmetric operator spaces associated with AFD factors of typeII and symmetric vector-valued spaces

If is a surjective isometry of the separable symmetric operator spaceE(M, ) associated with the approximately finite-dimensional semifinite factorM and if · _E(M,) is not proportional to · _{L 2}, then there exist a unitary operatorUM and a Jordan automorphismJ ofM such that(x)=UJ(x) for allxME(M, ). We characterize also surjective isometries of vector-valued symmetric spacesF((0, 1), E(M, )).Research supported by the Australian Research Council     

A constructive approach to the quantum (cosh ϕ)2 model. I. The method of the Gel'fand-Levitan-Marchenko equations

In a Hubert , with the aid of the postulated Gel'fand-Levitan-Marchenko quantum equations, one introduces the fields ₁(x) and ₂(x), which are the quantum analogues of the classical fields cosh (x) and sinh (x) in the sinh-Gordon model. It is shown that the fields _j(x) satisfy the Wightman axioms, including the invariance relative to reflections of space-time and mutual local commutativity. In addition, one proves the asymptotic completeness of the theory and one computes explicitly the scattering operator. In the developed approach, no cut-offs are used and, therefore, there are no renormalization effects.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 146, pp. 147–190, 1985.     

Best approximations of continuous functions by spline functions

An investigation of the approximation on [0, 1] of functionsf (x) by spline functions s(f,; x) of degree 2r-1 and of deficiency r (r>1) depending on the vector function = ₁ (x),..., _r-1(x) and interpolatingf (x) at fixed points. For the optimal choice of the vector ₀, exact estimates are obtained of the norms f(x)-s (f, ₀; x)_C[0,1] and f (x)-s (f, ₀; x)_{L[0, 1]} on the function classes H Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 41–46, July, 1970.In conclusion we would like to thank N. P. Korneichuk for suggesting this problem and for his valuable advice.     

On optimal extreme-discrepancy point sets in the square

Summary The optimal extreme-discrepancyN-point sets in the unit square are found forN6. The point ordering is unique forN3, but there are 14, 29 and 22 distinct orderings up to reflection about the diagonal forN=4, 5 and 6, respectively. The minimum discrepancy is less than, equal to, and greater thanN ^–1 forN3, 4N6 andN7, respectively.This work was performed at the University of Wisconsin with support from the National Science Foundation.     

A Class of Operators on L h 2 . II

Let D be the open unit disk in C, and L _h ² the space of quadratic integrable harmonic functions defined on D. Let be a function in L(D) with the property that (b) = lim _x b,xD (x) for all b D. Define the operator C on L _h ² as follows: C(f) = Q( · f), where Q is the orthogonal projection of L²(D) onto L _h ² . In this paper it is shown that if C is Fredholm, then is bounded away from zero on a neighborhood of D. Also, if C is compact, then |D 0, and the commutator ideal of (D) is K(D), where (D) denotes the norm closed subalgebra of the algebra of all bounded operators on L _h ² generated by , and K(D) is the ideal of compact operators on L _h ² . Finally, the spectrum of classes of operators defined on L _h ² is characterized.     

Spektrale Invarianz bei verallgemeinerten Eigenfunktionsentwicklungen

Let T be a selfadjoint operator in a Gelfand triplet H. Examples show that there can occur generalized eigenvalues of T which are not in the Hilbert space spectrum (T) of T. Moreover, the fuction d, which assigns to each real number s the dimension of the generalized eigenspace corresponding to s, can be essentially greater then the von Neumann multiplicity function of T. We therefore construct a new triplet H, closely related to the given Gelfand triplet, according to which the set of generalized eigenvalues of T is contained in (T), and the function d essentially equals the von Neumann multiplicity function of T. Then, in particular, the closure of the set of generalized eigenvalues equals (T). The expansion theorems in H are transferred to H.

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Grundlage dieser Arbeit ist ein Teil meiner Dissertation. Ich danke Herrn Prof. Dr. H.G. Tillmann für die Anregung hierzu und für viele wertvolle Hinweise.     

Local isometric immersions of two-dimensional metrics inE 4 with prescribed Gaussian torsion

Suppose that in a domain R(, B) of variables (r, ): (0 r , ₁ +B(r–r ₀ ) ₂–B(r–r₀), where > 0, B > 0, ₁ < 0 < ₂ are numbers) a metric ds² = dr² +G(r, )d ² and a function k(r, ) are given. The problem of isometrically immersing ds² in E ⁴ with prescribed Gaussian torsion is considered. The following is proved: The class C ⁵ metric ds ² is locally realized in the form of a class C ³ surface F ² whose Gaussian torsion is the prescribed class C ³ function (r, ).Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 38–47, 1992.     

Modified critical behavior in the ϕ4(O n ) model

In the standard ⁴(O _n ) model, a critical regime in which the coupling constantg of the ⁴ decreases as a certain given power as T-T _c0 is considered. From the point of view of physics, such a formulation of the problem corresponds to a certain class of trajectories of approach to the triple point in the two-dimensional plane of the physical parameters of the system. It is shown that in such a modified critical regime all the critical dimensions can be expressed in terms of the specified value of the exponent and the ordinary critical dimensions of the ⁴ model known in the form of 4- expansions.Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 1, pp. 168–172, April, 1992.     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.