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  • 1
    Electronic Resource
    Electronic Resource
    On monotone extensions of boundary data (1991)
    Springer
    Numerische Mathematik 60 (1991), S. 477-492 
    Details
    ISSN: 0945-3245
    Keywords: 65D15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A functionf ∈C (Ω), $$\Omega \subseteq \mathbb{R}^s $$ is called monotone on Ω if for anyx, y ∈ Ω the relation x − y ∈ ∝ + s impliesf(x)≧f(y). Given a domain $$\Omega \subseteq \mathbb{R}^s $$ with a continuous boundary ∂Ω and given any monotone functionf on ∂Ω we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of Ω and agree withf on ∂Ω. In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when Ω is the unit square [0, 1]2 in ∝2, strictly monotone analytic boundary data admit a monotone analytic extension.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385732
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  • 2
    Electronic Resource
    Electronic Resource
    Polynomial approximation inL p (0〈p〈1) (1992)
    Springer
    Constructive approximation 8 (1992), S. 187-201 
    Details
    ISSN: 1432-0940
    Keywords: 41A25 ; 41A20 ; Degree of approximation ; Monotone approximation ; Polynomials
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We prove that forf∈L p , 0〈p〈1, andk a positive integer, there exists an algebraic polynomialP n of degree ≤n such that $$\left\| {f - P_n } \right\|_p \leqslant C\omega _k^\varphi \left( {f,\frac{1}{n}} \right)_p $$ whereω k ϕ (f,t)p is the Ditzian-Totik modulus of smoothness off inL p , andC is a constant depending only onk andp. Moreover, iff is nondecreasing andk≤2, then the polynomialP n can also be taken to be nondecreasing.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01238268
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  • 3
    Electronic Resource
    Electronic Resource
    Preface (1993)
    Springer
    Constructive approximation 9 (1993), S. 121-121 
    Details
    ISSN: 1432-0940
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01198000
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  • 4
    Electronic Resource
    Electronic Resource
    On the construction of multivariate (pre)wavelets (1993)
    Springer
    Constructive approximation 9 (1993), S. 123-166 
    Details
    ISSN: 1432-0940
    Keywords: Primary 41A63 ; 46C99 ; Secondary 41A30 ; 41A15 ; 42B99 ; 46E20 ; Wavelets ; Multiresolution ; Shift-invariant spaces ; Box splines
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new approach for the construction of wavelets and prewavelets onR d from multiresolution is presented. The method uses only properties of shift-invariant spaces and orthogonal projectors fromL 2(R d ) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows a simple derivation of previous, as well as new, constructions of wavelets, and leads to a complete resolution of questions concerning the nature of the intersection and the union of a scale of spaces to be used in a multiresolution.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01198001
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  • 5
    Book
    Book
    Constructive approximation /; 303 (1993)
    Berlin [u.a.] :Springer,
    Details
    Title: Constructive approximation /; 303
    Author: DeVore, Ronald A.
    Contributer: Lorentz, George G.
    Publisher: Berlin [u.a.] :Springer,
    Year of publication: 1993
    Pages: X, 449 S. : , graph. Darst.
    Series Statement: ¬Die¬ Grundlehren der mathematischen Wissenschaften 303
    ISBN: 3-540-50627-6 , 0-387-50627-6
    Type of Medium: Book
    Language: English
    URL: http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0797.41016
    URL: lizenzfrei
    URL: 04
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