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  • 1
    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)
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  • 2
    Electronic Resource
    Electronic Resource
    Algorithms for cardinal interpolation using box splines and radial basis functions (1991)
    Springer
    Numerische Mathematik 60 (1991), S. 97-114 
    Details
    ISSN: 0945-3245
    Keywords: 65D05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We describe an algorithm for (bivariate) cardinal interpolation which can be applied to translates of “basis functions” which include box splines or radial basis functions. The algorithm is based on a representation of the Fourier transform of the fundamental interpolant, hence Fast Fourier Transform methods are available. In numerical tests the 4-directional box spline (transformed to the characteristical submodule of ℤ2), the thin plate spline, and the multiquadric case give comparably equal and good results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385716
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Multivariate Bernoulli splines and the periodic interpolation problem (1991)
    Springer
    Constructive approximation 7 (1991), S. 105-122 
    Details
    ISSN: 1432-0940
    Keywords: 41A15 ; 41A63 ; 46E20 ; 41A25 ; Bernoulli spline ; Periodic spline ; Reproducing kernel ; Spline interpolation ; Minimum norm interpolation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Periodic spline interpolation in Euclidian spaceR d is studied using translates of multivariate Bernoulli splines introduced in [25]. The interpolating polynomial spline functions are characterized by a minimal norm property among all interpolants in a Hilbert space of Sobolev type. The results follow from a relation between multivariate Bernoulli splines and the reproducing kernel of this Hilbert space. They apply to scattered data interpolation as well as to interpolation on a uniform grid. For bivariate three-directional Bernoulli splines the approximation order of the interpolants on a refined uniform mesh is computed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01888148
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    Paper (German National Licenses)
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