About some quadrature formulas

In this paper, we studied a class of quadrature formulas obtained by using the connection between the monospline functions and the quadrature formulas. For this class we obtain the optimal quadrature formula with regard to the error and we give some inequalities for the remainder term of this optimal quadrature formula.      

Error estimates for interpolatory quadrature formulae

Summary In this paper we study the remainder of interpolatory quadrature formulae. For this purpose we develop a simple but quite general comparison technique for linear functionals. Applied to quadrature formulae it allows to eliminate one of the nodes and to estimate the remainder of the old formula in terms of the new one. By repeated application we may compare with quadrature formulae having only a few nodes left or even no nodes at all. With the help of this method we obtain asymptotically best possible error bounds for the Clenshaw-Curtis quadrature and other Pólya type formulae.Our comparison technique can also be applied to the problem of definiteness, i.e. the question whether the remainderR[f] of a formula of orderm can be represented asc·f ^(m)(). By successive elimination of nodes we obtain a sequence of sufficient criteria for definiteness including all the criteria known to us as special cases.Finally we ask for good and worst quadrature formulae within certain classes. We shall see that amongst all quadrature formulae with positive coefficients and fixed orderm the Gauss type formulae are worst. Interpreted in terms of Peano kernels our theorem yields results on monosplines which may be of interest in themselves.     

The remainder term in quadrature formulae

Summary It is shown that the remainder term of any quadrature formula has an asymptotic expansion in terms of the step size; the occurrence of remainder terms of formc f ⁽ⁿ⁺¹⁾ () is discussed.     

The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegő type

We consider the Gauss-Kronrod quadrature formulae for the Bernstein-Szeg? weight functions consisting of any one of the four Chebyshev weights divided by the polynomial \(\rho (t)=1-\frac {4\gamma }{(1+\gamma )^{2}}\,t^{2},\quad t\in (-1,1),\ -1<\gamma \le 0\). For analytic functions, the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points ? 1 and sum of semi-axes ρ > 1, for the given quadrature formula. Starting from the explicit expression of the kernel, we determine the locations on the ellipses where maximum modulus of the kernel is attained. So we derive effective error bounds for this quadrature formula. An alternative approach, which has initiated this research, has been proposed by S. Notaris (Numer. Math. 103, 99–127, 2006).     

Distribution of zeta zeros and the oscillation of the error term of the prime number theorem

An 84-year-old classical result of Ingham states that a rather general zero-free region of the Riemann zeta function implies an upper bound for the absolute value of the remainder term of the prime number theorem. In 1950 Tur´an proved a partial conversion of the mentioned theorem of Ingham. Later the author proved sharper forms of both Ingham’s theorem and its conversion by Tur´an. The present work shows a very general theorem which describes the average and the maximal order of the error terms by a relatively simple function of the distribution of the zeta zeros. It is proved that the maximal term in the explicit formula of the remainder term coincides with high accuracy with the average and maximal order of the error term.     

一类Gauss-Jacobi数值求积公式的极限性质

讨论了形如∫aa+h(x-a)βf(x)dx的Gauss-Jacobi求积公式,当积分区间长度趋向于零时,确定了求积公式的余项中介点η的渐近性,并给出了校正公式,比原公式提高了两次代数精度.此外,本文的结论包含了文[3]的结果.     

A characterization of the entropy functionals for grand canonical ensembles. The discrete case

Summary ? Natural ? properties lead to a representation theorem for the entropy functional of a grand canonical ensemble. In addition to the classical Boltzmann term, the representation contains three more terms that seem to be meaningful in statistical mechanics.

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Sunto Da proprietà ? naturali ? imposte al funzionale entropia di un insieme gran canonico si deduce un teorema di rappresentazione per esso. In aggiunta al termine classico di Boltzmann, la rappresentazione mette in luce tre ulteriori termini che sembrano significativi in meccanica statistica.

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A Dario Graffi per il suo 70° compleanno

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Entrata in Redazione il 17 marzo 1976.

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While on sabbatical leave of absence from the University of Waterloo, Canada, Department of Applied Mathematics.

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Research supported in part by National Research Council of Canada, grant A-7677 and IBM-Italia.     

Ein einfacher Beweis des Satzes von Alexandroff-Lester

By presenting counterexamples and a new proof, we determine all metric vector spaces, for which the theorem of ALEXANDROFF-LESTER holds. In this context, the theorem will be extended to certain nonregular metric vector spaces of characteristic 2.

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Dedicated to Professor Giuseppe Tallini on the occasion of his 60th birthday

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Herrn H. Lenz danke ich für mehrere Vorschläge zur durchsichtigeren Formulierung der Beweise.     

Un teorema di esistenza per controlli ottimi di sistemi definiti da equazioni integrali di Urysohn

Summary In this paper an existence theorem is proved for optimal control problems described by Urysohn systems, with pointwise constraints on controls and states. An example of application of the existence theorem is given.

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Entrata in Redazione il 15 marzo 1978.

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Lavoro eseguito nell'ambito della mia attività di ricerca come borsista C.N.R., presso l'Università di Genova.     

Der ĉebotarev’sche dichtigkeitssatz und ein analogon zum dirichlet’schen primzahlsatz für algebraische funktionenkörper

We improve the remainder term in Čebotarev’s density theorem for algebraic function fields by a logarithmic factor. With aid of this, we deduce an analogon of Dirichlet’s theorem on primes in arithmetic progressions for holomorphy rings of algebraic function fields with best possible remainder term.      

On asymptotic linearity of L-estimates

A theorem on asymptotic linearity of L-estimates is proved under general set of regularity conditions, allowing the sampled distribution to be non-integrable. The main result is the improvement in the order of the remainder term in the formula for asymptotic linearity of L-statistic. It is shown that in the case of the integral coefficients this term R _n = O _P (1/n) and the case of functional coefficients is also covered.     

Error estimates for Gaussian quadratures of analytic functions

For analytic functions the remainder term of Gaussian quadrature formula and its Kronrod extension can be represented as a contour integral with a complex kernel. We study these kernels on elliptic contours with foci at the points ±1 and the sum of semi-axes ?>1 for the Chebyshev weight functions of the first, second and third kind, and derive representation of their difference. Using this representation and following Kronrod’s method of obtaining a practical error estimate in numerical integration, we derive new error estimates for Gaussian quadratures.     

Sul teorema di Riesz-Thorin

Summary We give an extension of the Riesz-Thorin theorem to linear operators which map Morrey spaces in spaces of the same or similar type.

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A Bruno Finzi nel suo 70^mo compleann

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Entrata in Redazione il 10 dicembre 1969.     

Ein Beweis von H. Karzel und I. Pieper

An incidence space of dimension at least 2 and order at least 4 in which all planes are affine planes is an affine space ([1 ] and [2]). Karzel's and Pieper's proof of this theorem is represented here lightened from some material relevant to the report [2] but not to this theorem.

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Herrn Professor Dr. Adriano Barlotti zum 65. Geburtstag     

Estimating quadrature errors for analytic functions using kernel representations and biorthogonal systems

Summary. For analytic functions the remainder term of quadrature rules can be represented as a contour integral with a complex kernel function. From this representation different remainder term estimates involving the kernel are obtained. It is studied in detail how polynomial biorthogonal systems can be applied to derive sharp bounds for the kernel function. It is shown that these bounds are practical to use and can easily be computed. Finally, various numerical examples are presented. Received March 11, 1998 / Revised version January 22, 1999/ Published online November 17, 1999     

Ein funktionalanalytischer Beweis des Satzes von Cauchy-Kowalewsky

Using functional analytic methods, in particular the fixed point theorem of Schauder-Tychonoff, a new and short proof is given for the classical Cauchy-Kowalewsky theorem on analytic partial differential equations.

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Herrn Prof. Dr. F.W. Schafke zum 60. Geburtstag gewidmet     

Fehlerabschätzung für die numerische Differentiation analytischer Funktionen durch Quadratur

Summary Numerical treatment of the integral in Cauchy's integral formula produces approximations for the derivatives of an analytic functionf; this fact has already been utilized byLyness andMoler [3, 4]. In the present paper this idea is investigated especially in view of the accuracy of these formulas regarded as quadrature formulas. Since the integration can be reduced to the integration of a periodic analytic function, it is possible to continue the considerations ofDavis [2] in order to find bounds for the error of the differentiation rules. For the application of these bounds one essentially needs estimations of the maximum off on a circle inside of its region of analyticity. Examples show the practical use of the bounds.

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Meinem verehrten LehrerH. Görtler zur Vollendung seines 60. Lebensjahres gewidmet     

Eine Bemerkung über den speziellen Bezoutschen Satz

Some synthetic methods in a projective space are devellopped and using them a theorem connected with the theorem of Bézout is proved without using any postulation about algebraic closure.

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Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag     

Un teorema di esistenza per campi di orientori e applicazioni al problema libero del Calcolo delle Variazioni in dimensione infinita

Summary We prove a compactness theorem for a sequence of functions which take values in an infinite dimensional Hilbert space and which satisfy a differential inclusion, related to a map Q. Then we give sufficient conditions for the existence of an optimal solution of a free Lagrange problem, always in the infinite dimensional case, applying the theorem previously proved to a minimizing sequence.

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Entrata in Redazione il 19 maggio 1978.     

On the group of the automorphisms of some algebraic systems

Summary Within a framework of general algebra we firstly formulate a proposition on the group of the automorphisms of some irreducible algebrae (id est algebrae without proper non trivial subalgebrae). This proposition includes as particular cases the uniqueness of the automorphisms of the rational field and the Burnside theorem on the commutant of an irreducible set of operators of a finite dimensional vector space over an algebraically closed field. Afterwards we apply the general proposition to modules with irreducible sets of semilinear operators and we obtain a theorem which generalises from several points of view the Burnside theorem. Finally we derive as an application a proposition which specifies the set of the Hermitian operators that commute with an irreducible set of semilinear operators of a finite dimensional real, complex or quaternion scalar product space.

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Riassunto In un contesto di algebra generale si formula in primo lucgo una proposizione sul gruppo degli automorfismi di talune algebre irriducibili (ossia prive di sottoalgebre non banali). Questa proposizione comprende, come casi particolari, l'unicità degli automorfismi del campo razionale, e il teorema di Burnside sul commutante di un insieme irriducibile di operatori di uno spazio lineare di dimensione finita costruito su un campo algebricamente chiuso. Quindi si applica la proposizione generale a moduli con insiemi irriducibili di operatori semilineari e si ottiene un teorema che generalizza sotto diversi punti di vista il teorema di Burnside. Infine, si deduce, come applicazione, una proposizione che specifica l'insieme degli operatori hermitiani che commutano con un insieme irriducibile di operatori semilineari di uno spazio con prodotto scalare, reale, complesso o quaternionico.

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This work was partly supported at the Istituto di Fisica dell'Università di Torino by USAF EOAR Grant n. 68-0015.

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This work was partly performed at the Istituto di Matematica dell'Università di Palermo within the Group 45 of the Comitato Nazionale per le Scienze Matematiche del C.N.R.