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  • 1
    Electronic Resource
    Electronic Resource
    Error estimates for interpolatory quadrature formulae (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 371-386 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D30 ; CR: 5.16
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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)(ξ). 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400316
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  • 2
    Electronic Resource
    Electronic Resource
    Converse theorems in the theory of approximate integration (1990)
    Springer
    Constructive approximation 6 (1990), S. 321-334 
    Details
    ISSN: 1432-0940
    Keywords: Quadrature formula ; Entire functions of exponential type ; Fourier transform ; Paley-Wiener theorem ; Functions holomorphic in a strip ; Sobolev space
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We characterize certain function classes in terms of the remainder in the quadrature formula $$\int_{ - \infty }^\infty {f(x)dx = (\pi /\sigma )\sum\nolimits_{v = - \infty }^\infty {f(v\pi /\sigma ) + R_\sigma [f]} } $$ . In the process, we prove a generalization of the famous theorem of Paley and Wiener about entire functions of exponential type belonging toL 2 on the real line.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01890414
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  • 3
    Electronic Resource
    Electronic Resource
    The Sampling Theorem and Several Equivalent Results in Analysis (2000)
    Springer
    Journal of computational analysis and applications 2 (2000), S. 333-371 
    Details
    ISSN: 1572-9206
    Keywords: sampling theorem ; Cauchy's integral formula ; Poisson's summation formula ; Fourier analysis ; complex analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract First we show that several fundamental results on functions from theBernstein spaces $$B_\sigma ^p $$ (such as Bernstein's inequality andthe reproducing formula) can be deduced from a weak form of the classicalsampling theorem. In §3 we discuss the mutual equivalence of thesampling theorem, the derivative sampling theorem and a harmonic functionsampling theorem. In §§4–6 we discuss connections between thesampling theorem and various important results in complex analysis andFourier analysis. Our considerations include Cauchy's integral formula,Poisson's summation formula, a Gaussian integral, certain properties ofweighted Hermite polynomials, Plancherel's theorem, the maximum modulusprinciple, and the Phragmén–Lindelöf principle.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1010164717587
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  • 4
    Electronic Resource
    Electronic Resource
    On a phenomenon concerning (0,2)-interpolation (1978)
    Springer
    Periodica mathematica Hungarica 9 (1978), S. 163-171 
    Details
    ISSN: 1588-2829
    Keywords: Primary 41A05 ; Secondary 26A75 ; 30A06 ; (0, 2)-interpolation ; fundamental polynomials ; convergence behaviour
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02018931
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    Paper (German National Licenses)
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