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  • 1
    Electronic Resource
    Electronic Resource
    Regularity of multiwavelets (1997)
    Springer
    Advances in computational mathematics 7 (1997), S. 455-545 
    Details
    ISSN: 1572-9044
    Keywords: subdivision ; refinable functions ; regularity ; multiwavelets ; 39B12 ; 41A15 ; 41A25 ; 65D99
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The motivation for this paper is an interesting observation made by Plonka concerning the factorization of the matrix symbol associated with the refinement equation for B-splines with equally spaced multiple knots at integers and subsequent developments which relate this factorization to regularity of refinable vector fields over the real line. Our intention is to contribute to this train of ideas which is partially driven by the importance of refinable vector fields in the construction of multiwavelets. The use of subdivision methods will allow us to consider the problem almost entirely in the spatial domain and leads to exact characterizations of differentiability and Hölder regularity in arbitrary L p spaces. We first study the close relationship between vector subdivision schemes and a generalized notion of scalar subdivision schemes based on bi-infinite matrices with certain periodicity properties. For the latter type of subdivision scheme we will derive criteria for convergence and Hölder regularity of the limit function, which mainly depend on the spectral radius of a bi-infinite matrix induced by the subdivision operator, and we will show that differentiability of the limit functions can be characterized by factorization properties of the subdivision operator. By switching back to vector subdivision we will transfer these results to refinable vectors fields and obtain characterizations of regularity by factorization and spectral radius properties of the symbol associated to the refinable vector field. Finally, we point out how multiwavelets can be generated from orthonormal refinable bi-infinite vector fields.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018971524949
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  • 2
    Electronic Resource
    Electronic Resource
    H-bases for polynomial interpolation and system solving (2000)
    Springer
    Advances in computational mathematics 12 (2000), S. 335-362 
    Details
    ISSN: 1572-9044
    Keywords: ideal bases ; Gröbner bases ; multivariate polynomials ; interpolation ; systems of polynomial equations ; 65D05 ; 65H10 ; 13P10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The H-basis concept allows, similarly to the Gröbner basis concept, a reformulation of nonlinear problems in terms of linear algebra. We exhibit parallels of the two concepts, show properties of H-bases, discuss their construction and uniqueness questions, and prove that n polynomials in n variables are, under mild conditions, already H-bases. We apply H-bases to the solution of polynomial systems by the eigenmethod and to multivariate interpolation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018937723499
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Computational aspects of multivariate polynomial interpolation (1995)
    Springer
    Advances in computational mathematics 3 (1995), S. 219-237 
    Details
    ISSN: 1572-9044
    Keywords: Lagrange interpolation ; finite difference ; algorithm ; Primary ; 41A05 ; 41A10 ; 65D05 ; 65D10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The paper is concerned with the practical implementation of two methods to compute the solution of polynomial interpolation problems. In addition to a description of the implementation, practical results and several improvements will be discussed, focusing on speed and robustness of the algorithms under consideration.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02432000
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    Paper (German National Licenses)
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