On the multi-level splitting of finite element spaces

Summary In this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases. We show that the condition number of such a stiffness matrix behaves like O((log )²) where is the condition number of the stiffness matrix with respect to a nodal basis. In the case of a triangulation with uniform mesh sizeh this means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving like instead of for a nodal basis. The proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization. Especially we do not need the quasiuniformity of the employed triangulations. As the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis, our results mean that the method of conjugate gradients needs onlyO(log n) steps andO(n log n) computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures. Heren denotes the dimension of the finite element space and of the discrete linear problem to be solved.     

On the convergence of powers of interval matrices (2)

Summary Let be a real irreduciblen×n interval matrix. Then a necessary and sufficient condition is given for the sequence of the powers of an interval matrix to converge to a matrix which is not the null matrix. In addition a criterion for is proved to decide whether the limit matrix satisfies the condition of symmetry .     

Resonant nonlinear singular problems in the limit circle case

Existence results are presented for the resonant singular boundary value problem a.e. on [0, 1] with lim _t0+py=y(1)=0. Here we donot assume but only that .     

Exact asymptotic behavior of singular values of a class of integral operators

We find an exact asymptotic formula for the singular values of the integral operator of the form , a Jordan measurable set) where and L is slowly varying function with some additional properties. The formula is an explicit expression in terms of L and T.     

Stationary Harry-Dym's equation and its relation with geodesics on ellipsoid

Harry-Dym's equation (HD) is well-known for its cusp soliton solutions. In this paper, relations are revealed between HD and a completely integrable Hamiltonian system in Liouville sense given by

On a problem connected with zeros of ζ(s) on the critical line

The existence of zeros ofZ ^(k)(t) in short intervals of the type [T, T+H] is established, whereHT ^a(k)logT, . Hitherto the sharpest bounds for the constanta(k) are obtained by employing a certain exponential averaging technique and the estimation of the relevant exponential sums. Bounds for are also derived, under the assumption that orZ(t) does not vanish in certain short intervals.     

Integrals involving products of bessel functions

Summary The integrals and , where n is any positive integer, are evaluated in terms ofMacRobert E-functions and generalized hypergeometric functions.     

Truncation error bounds for modified continued fractions with applications to special functions

Much recent work has been done to investigate convergence of modified continued fractions (MCF's), following the proof by Thron and Waadeland [35] in 1980 that a limit-periodic MCFK(a _n , 1;x ₁), with andnth approximant

On the behavior of the solution of a stochastic equation with unbounded drift

We study the behavior as 0 of the solution of the equation with periodic coefficients

Das asymptotische Verhalten der Peanokerne einiger interpolatorischer Quadraturverfahren

Summary Interpolatory quadrature formulae consist in replacing by wherep _f denotes the interpolating polynomial off with respect to a certain knot setX. The remainder may in many cases be written as wherem=n resp. (n+1) forn even and odd, respectively. We determine the asymptotic behaviour of the Peano kernelP _X (t) forn for the quadrature formulae of Filippi, Polya and Clenshaw-Curtis.

     

A sharper stability bound of Fourier frames

Given a real sequence {_n}_n. Suppose that is a frame for L²[–, ] with bounds A, B. The problem is to find a positive constant L such that for any real sequence {_n}_n with ¦_n –_n¦ <L, is also a frame for L²[–, ]. Balan [1] obtained arcsin . This value is a good stability bound of Fourier frames because it covers Kadec's 1/4-theorem and is better than (see Duffin and Schaefer [3]). In this paper, a sharper estimate is given.     

Nikolskii type inequality for cardinal splines

The Nikolskii type inequality for cardinal splines

Zur Interpolation und Integration differenzierbarer periodischer Funktionen

Summary Letx ₀<x ₁<...<x _n–1<x ₀+2 be nodes having multiplicitiesv ₀,...,v _n–1, 1v _k r (0k<n). We approximate the evaluation functional ,x fixed, and the integral respectively by linear functionals of the form and determine optimal weights for the Favard classesW _r C ₂. In the even case of optimal interpolation these weights are unique except forr=1,x(x _k +x _k–1)/2 mod 2. Moreover we get periodic polynomial splinesw _{k, j} (0k<n, 0j<v _k ) of orderr such that are the optimal weights. Certain optimal quadrature formulas are shown to be of interpolatory type with respect to these splines. For the odd case of optimal interpolation we merely have obtained a partial solution.

Bojanov hat in [4, 5] ähnliche Resultate wie wir erzielt. Um Wiederholungen zu vermeiden, werden Resultate, deren Beweise man bereits in [4, 5] findet, nur zitiert     

About existence and uniqueness of the restricted total least squares estimation

Summary Total Least Squares (TLS) is an estimation method for the solutiona of the linear system when both data sets and are subject to error. The TLS-method minimizes the functional with weighting parameter . In this paper the TLS-functional is analyzed by the technique of Lagrangian multipliers. The main part of the work deals with the case when the estimatea is restricted by an inequality of the formD a–b0, D a diagonal matrix. It is shown that there exists a unique estimatea if the weighting parameter is chosen sufficiently large.     

Un nouveau type d'inégalités pour les martingales discrètes

Summary We prove the following extension of classical Burkholder-Davis-Gundy inequalities: let (X _n ) _nN be a martingale; for p1, in order that and belong to L ^p, it is sufficient that Inf(X ^*, S(X)) belong to L ^p. For «regular» martingales this result holds for p>0.     

The many limits of mixed means

The sequences introduced by Carlson (1971) are variants of the Gauss arithmetic geometric sequences (which have been elegantly discussed by D. A. Cox (1984, 1985)). Given (complex)a ₀,b ₀ we define

On incomplete Gaussian sums

In this paper it is proved that

Monotony in interpolatory quadratures

Summary Let , be holomorphic in an open disc with the centrez ₀=0 and radiusr>1. LetQ _n (n=1, 2, ...) be interpolatory quadrature formulas approximating the integral . In this paper some classes of interpolatory quadratures are considered, which are based on the zeros of orthogonal polynomials corresponding to an even weight function. It is shown that the sequencesQ _n 9f] (n=1, 2, ...) are monotone. Especially we will prove monotony in Filippi's quadrature rule and with an additional assumption onf monotony in the Clenshaw-Curtis quadrature rule.     

Product formulas and convolution structure for Fourier-Bessel series

One of the most far-reaching qualities of an orthogonal system is the presence of an explicit product formula. It can be utilized to establish a convolution structure and hence is essential for the harmonic analysis of the corresponding orthogonal expansion. As yet a convolution structure for Fourier-Bessel series is unknown, maybe in view of the unpractical nature of the corresponding expanding functions called Fourier-Bessel functions. It is shown in this paper that for the half-integral values of the parameter ,n=0, 1, 2,, the Fourier-Bessel functions possess a product formula, the kernel of which splits up into two different parts. While the first part is still the well-known kernel of Sonine's product formula of Bessel functions, the second part is new and reflects the boundary constraints of the Fourier-Bessel differential equation. It is given, essentially, as a finite sum over triple products of Bessel polynomials. The representation is explicit up to coefficients which are calculated here for the first two nontrivial cases and . As a consequence, a positive convolution structure is established for . The method of proof is based on solving a hyperbolic initial boundary value problem.Communicated by Tom H. Koornwinder.     

An Ω-theorem for an error term related to the sum-of-divisors function

Let denote the sum-of-divisors function, and set . Gronwall and Wigert proved (independently) in 1913 and 1914, respectively, thatE ₁ (x)= (x log logx). In this paper we obtain the more preciseE ₁ (x)=_–(x log logx). The method consists in averaging over suitable arithmetic progressions, and was suggested by the work ofP. Erdös andH. N. Shapiro [Canad. J. Math. 3–4, 375–385 (1951)] on the error term corresponding to Euler's functions, .