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  • 41 A 20  (1)
  • 42B15  (1)
  • AMS(MOS): 41A15, 41A25, 41A63  (1)
  • 1
    Electronic Resource
    Electronic Resource
    A characterization of multivariate quasi-interpolation formulas and its applications (1990)
    Springer
    Numerische Mathematik 57 (1990), S. 105-121 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 41A15, 41A25, 41A63
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Let ϕ be a compactly supported function on ℝ s andS (ϕ) the linear space withgenerator ϕ; that is,S (ϕ) is the linear span of the multiinteger translates of ϕ. It is well known that corresponding to a generator ϕ there are infinitely many quasi-interpolation formulas. A characterization of these formulas is presented which allows for their direct calculation in a variety of forms suitable to particular applications, and in addition, provides a clear formulation of the difficult problem of minimally supported quasi-interpolants. We introduce a generalization of interpolation called μ-interpolation and a notion of higher order quasi-interpolation called μ-approximation. A characterization of μ-approximants similar to that of quasi-interpolants is obtained with similar applications. Among these applications are estimating least-squares approximants without matrix inversion, surface fitting to incomplete or semi-scattered discrete data, and constructing generators with one-point quasi-interpolation formulas. It will be seen that the exact values of the generator ϕ at the multi-integers ℤ s facilitates the above study. An algorithm to yield this information for box splines is discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01386401
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Order of approximation by electrostatic fields due to electrons (1985)
    Springer
    Constructive approximation 1 (1985), S. 121-135 
    Details
    ISSN: 1432-0940
    Keywords: 41 A 20 ; 30 E 10 ; Order of approximation ; Rational functions and electrostatic fields due to distribution of electrons ; Bers space ; Jordan domain ; Conformal mapping
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract LetD be a Jordan domain in the complex plane andA q (D) the Bers space with norm ∥ ∥ q . IfD is the unit disk, it is known that ∥S n 0∥2≥π/18, whereS n =∑ k=1 n l/(z−z nk ) withz nk ∈∂D, so that approximation in ∥ ∥ q ,q〈-2, is not possible. In this paper, we give an order of estimate of ∥f−S n ∥ q for 2〈q〈∞ when ∂D is a sufficiently smooth Jordan curve, and prove that this order of approximation is in general best possible.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01890026
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Affine frames, quasi-affine frames, and their duals (1998)
    Springer
    Advances in computational mathematics 8 (1998), S. 1-17 
    Details
    ISSN: 1572-9044
    Keywords: affine frame ; quasi-affine frame ; dual ; sesquilinear operator ; translation invariance ; 42C15 (primary) ; 42B15 ; 47N40 (secondary)
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The notion of quasi-affine frame was recently introduced by Ron and Shen [9] in order to achieve shift-invariance of the discrete wavelet transform. In this paper, we establish a duality-preservation theorem for quasi-affine frames. Furthermore, the preservation of frame bounds when changing an affine frame to a quasi-affine frame is shown to hold without the decay assumptions in [9]. Our consideration leads naturally to the study of certain sesquilinear operators which are defined by two affine systems. The translation-invariance of such operators is characterized in terms of certain intrinsic properties of the two affine systems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018975725857
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    Paper (German National Licenses)
    Fulltext
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