An error analysis for absolute and relative approximation

We consider the vectorial algorithm for finding best polynomial approximationsp P _n to a given functionf C[a, b], with respect to the norm · _s , defined byp – f _s =w ₁ (p – f)+w ₂ (p – f) A bound for the modulus of continuity of the best vectorial approximation operator is given, and using the floating point calculus of J. H. Wilkinson, a bound for the rounding error in the algorithm is derived. For givenf, these estimates provide an indication of the conditioning of the problem, an estimate of the obtainable accuracy, and a practical method for terminating the iteration.This paper was supported in part by the Canadian NCR A-8108, FCAC 74-09 and G.E.T.M.A.Part of this research was done during the first-named author's visit to theB! Chair of Applied Mathematics, University of Athens, Spring term, 1975.     

Error bounds for a near-minimax approximation

It is known that a near minimax polynomial approximation tof C [–1, 1] is provided by a finite carrier projectionM _n fromC[–1, 1] onto the subspace of all polynomials of degree n, such thatM _nf is a weighted least squares approximation off on the set consisting of the extreme points of the Chebyshev polynomialT _{2n + 1}. In this paper, upper bounds for the error f–M _n f are given in terms of divided differences.     

Approximating derivations on ideals ofC *-algebras

For each^*-derivation of a separableC ^*-algebraA and each >0 there is an essential idealI ofA and a self-adjoint multiplierx ofI such that (–ad(ix))|I< and x.     

On the Sobolev distance of convex bodies

Summary LetK ^d denote the cone of all convex bodies in the Euclidean spaceK ^d . The mappingK h _K of each bodyK K ^d onto its support function induces a metric _w onK ^d by" _w (K, L)h _L –h _{K w} where _w is the Sobolev I-norm on the unit sphere . We call _w (K, L) the Sobolev distance ofK andL. The goal of our paper is to develop some fundamental properties of the Sobolev distance.     

Interpolating them-th power ofx at the zeros of then-th Chebyshev-polynomial yields an almost best Chebyshev-approximation

LetX={x ₁,x ₂,..., _n }I=[–1, 1] and . ForfC ₁(I) definef* byf–p _f =f*, wherep _f denotes the interpolation-polynomial off with respect toX. We state some properties of the operatorf f*. In particular, we treat the case whereX consists of the zeros of the Chebyshev polynomialT _n (x) and obtain x ^m –p _x ^m8eE _n–1(x ^m ), whereE _n–1(f) denotes the sup-norm distance fromf to the polynomials of degree less thann. Finally we state a lower estimate forE _n (f) that omits theassumptionf ⁽ⁿ⁺¹⁾>0 in a similar estimate of Meinardus.     

Two counterexamples related to the Kreiss matrix theorem

The modern version of the Kreiss matrix theorem states that A ⁿ esK (for alln0) ifA is ans ×s matrix satisfying a resolvent condition with respect to the unit disk with constantK. In this paper we show for any fixedK + 1 that the upper boundesK is sharp in the sense that a linear dependence on the dimensions is the best possible. An analogous result is obtained for the continuous version of the Kreiss matrix theorem, which states that exp(tA) esK (for allt0) ifA is ans ×s matrix satisfying a resolvent condition with respect to the left half plane with constantK.The research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences (K.N.A.W.) and was carried out at the Department of Mathematics and Computer Science, University of Leiden, The Netherlands.     

Jordan axioms for C*-algebras

A complex Banach spaceA which is also an associative algebra provided with a conjugate linear vector space involution * satisfying (a ²)^*=(a ^*)², aa ^* a=a³ and ab+ba2ab for alla, b inA is shown to be a C^*-algebra. The assumptions onA can be expressed in terms of the Jordan algebra obtained by symmetrization of the product ofA and are satisfied by any C^*-algebra. Thus we obtain a purely Jordan characterization of C^*-algebras.     

On the convergence of multivariable Lagrange interpolants

LetA _dj be a triangular array in a compact setXC ⁿ . Forf analytic in a neighborhood ofX, letL _d (f) denote the Lagrange interpolant tof at staged of the array. In the caseX is locally regular, we construct a continuous function satisfying the complex Monge-Ampère equation onC ⁿ –X, such that iff is analytic onR forR>1 then, for someB>0, we have L _d (f)–f _x B exp(–d logR. In particular, since1 onX, iff is analytic on1, then lim _d L _d (f)–f _x =0.Communicated by Edward B. Saff.     

On the rank of a spectral function

Let P(x), 0 x 1, be an absolutely continuous spectral function in the separable Hilbert spacesS. If the vectors h_j, j=1, 2, ..., s; s are such that the set P(x)h_j is complete inS, then the rank of the function P(x) equals the general rank of the matrix-function d/dxP(x)h_i,h_js₁.Translated from Matematicheskie Zametki, Vol. 5, No. 4, pp. 457–460, April, 1969.     

Ein neues Surjektivitätskriterium im Hilbertraum

LetA, B andC be linearm-accretive operators in a Hilbert space. Suppose further thatC is bounded, thatb:=inf {Re (C y, y)| y=1}>0, thatA ^–1 exists as a bounded operator and that Re (B ^* x, A ^–1 x)+a x²0 holds for allxD (B ^*) and a constanta with 0a<b. ThenCA+B is surjective, (CA+B)^–1 exists and C Ax+Bx (b–a) A x holds for allxD (A) D (B). This criterion can be applied to evolution equations of the formdu/dt+C(t)A(t)u=f(t) whereB:=d/dt.     

Über die Existenz gemeinsamer Elemente bester Approximation bezüglich zweier Normen II

The present paper deals with the possibility of existence of best approximation elements, simultaneously with respect to two norms ·_i,i=1,2, for all the elements of a class of subspaces. In case this class in any of the following: (a) All n-dimensional subspaces, (b) All ·₁-or ·||₂-closed, n-codimensional subspaces, (c) All ·₁-or ·₂-closed subspaces with infinite dimension and codimension, we prove that the two norms differ at most by a constant factor.     

Exact bounds for norms of differentiable periodic functions in L metric

Exact values are obtained for the upper bounds on the norms,fL, on the classes W ^(r) H(r =0, 1, 2,...) of r-fold differentiable functions f(x), of periodicity2, for which (f^(r);t)(t),where (t) is a given convex modulus of continuity.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 569–576, December, 1967.     

Problem of instability in the first approximation

Let E be a Banach space, A be a continuous linear operator such that (A) ; Re>0 Ø, and F(t, x) be a continuous function on [0, )×E satisfying the condition F(t, x) q x (q= const). An example of a system dx/dt=Ax + F(t, x) is given which has an exponentially stable zero solution for certain F(t, x) with arbitrarily small q.Translated from Matematicheskie Zametki, Vol. 23, No. 5, pp. 721–723, May, 1978.     

Absorption spectroscopy and excitation of YAlO3∶Er (50%) crystals

Absorption spectra have been measured in highly doped YAlO₃Er (50 %) in the wavelength range from 300 nm up to 2m. The spectra were taken with irradiation to the crystallographic b-axis with polarization a- or c-axis. From the measured spectra the Er energy levels have been determied. Based on the results of absorption spectroscopy the pumplight absorption in a Xenon-flashlamp-pumped YAlO₃Er laser slab has been calculated as a function of wavelength and depth.

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Zusammenfassung In hochdotiertem YAlO₃Er (50%) wurden Absorptionsspektren im Wellenlängenbereich von 300 nm bis 2m gemessen. Die Spektren wurden mit Bestrahlung zu der kristallographischen b-Achse mit Polarisation zur a- oder c-Achse aufgenommen. Aus den gemessenen Spektren wurden die Energieniveaus bestimmt. Mit den Ergebnissen aus der Absorptionsspektroskopie wurde die Pumplichtabsorption in einer mit Xenonblitzlampen gepumpten YAlO₃Er Platte in Abhängigkeit von der Wellenlänge berechnet.

     

Sufficient conditions for bang-bang control in Hilbert space

Sufficient conditions for bang-bang and singular optimal control are established in the case of linear operator equations with cost functionals which are the sum of linear and quadratic terms, that is,Ax=u,J(u)=(r,x)+(x,x), >0. For example, ifA is a bounded operator with a bounded inverse from a Hilbert spaceH into itself and the control setU is the unit ball inH, then an optimal control is bang-bang (has norm l) if 0<1/2;A ^–1*r·A ^–1–2, but is singular (an interior point ofU) if >1/2A ^–1*r·A².This work was supported by NRC Grant No. A-4047 and NSF Grant No. GP-7445.     

Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization

Let be a centered Gaussian measure on a separable Hilbert space (E, ). We are concerned with the logarithmic small ball probabilities around a -distributed center X. It turns out that the asymptotic behavior of –log (B(X,)) is a.s. equivalent to that of a deterministic function _R (). These new insights will be used to derive the precise asymptotics of a random quantization problem which was introduced in a former article by Dereich, Fehringer, Matoussi, and Scheutzow.⁽⁸⁾     

Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

For the classB _p , 0 < 1, 1p , of 2-periodic functions of the form f(t)=u(,t), whereu (,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K(t) of the convolution f= K *g, gl, with respect to the metric of L₁. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1549–1557, November, 1995.     

Joint norm of operators and an investigation of nonlinear problems

Nonlinear operator equations of the form x=Fx in a real-valued Hilbert space H are studied. If the operator F is completely continuous and admits the bound Fx< Bx+b, where B is a continuous linear operator then for B<1 the Schauder principle is applicable to the equation x=Fx and this equation possesses at least one solution x H. If the bound Fx<,B₁x+B₂x+b is valid where B₁ and B₂ are bounded linear operators then the simplest conditions for solvability of the equation x=Fx is of the form B₁+B₂<1. This condition could be relaxed. The proposed method is applied to the investigation of a two-point boundary problem (cf., e.g., [1–3]). New conditions for the existence of solutions are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1605–1616, December, 1990.     

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     

Isolation theorem for forms corresponding to purely real algebraic fields

Let M be the complete module of a purely real algebraic field of degree n 3, let be a lattice in this module, and let F(X) be its form. We use to denote any lattice for which we have = , where is a nondiagonal matrix for which – I . With each lattice we can associate a factorizable formF (X) in a natural manner. We denote the complete set of forms corresponding to the set {} by {F (X)}. It is proved that for any > 0 there exists an > 0 such that for eachF (X) {F } we have |F (X₀₎| for some integer vector X₀ 0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 5–12, 1990.In conclusion, the author would like to express his deep gratitude to B. F. Skubenko for stating the problem and for his constant attention.