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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Constructive approximation 3 (1987), S. 239-248 
    ISSN: 1432-0940
    Keywords: Multivariate approximation ; Splines with free knots ; Order of approximation ; Besov spaces ; 41A20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We introduce a definition of free multivariate splines which generalizes the univariate notion of splines with free knots. We then concentrate on the simplest case, piecewise constant functions and characterize some classes of functions which have a prescribed order of approximation inL p by these splines. These characterizations involve the classical Besov spaces.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Constructive approximation 8 (1992), S. 187-201 
    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.
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Constructive approximation 9 (1993), S. 121-121 
    ISSN: 1432-0940
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Constructive approximation 1 (1985), S. 323-331 
    ISSN: 1432-0940
    Keywords: 41 A 10 ; 41 A 25 ; 41 A 29 ; Monotone approximation by algebraic polynomials ; Degree of approximation ; Pointwise estimates
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We prove that iff is increasing on [−1,1], then for eachn=1,2,... there is an increasing algebraic polynomialP n of degreen such that |f(x)−P n (x)|≤cω 2(f,√1−x 2/n), whereω 2 is the second-order modulus of smoothness. These results complement the classical pointwise estimates of the same type for unconstrained polynomial approximation. Using these results, we characterize the monotone functions in the generalized Lipschitz spaces through their approximation properties.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Constructive approximation 9 (1993), S. 123-166 
    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
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 60 (1991), S. 477-492 
    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.
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Manuscripta mathematica 63 (1989), S. 469-478 
    ISSN: 1432-1785
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We introduce a definition of nonlinear n-widths and then determine the n-widths of the unit ball of the Sobolev spaceW p r inL q. We prove that in the sense of these widths the manifold of splines of fixed degree with n free knots is optimal for approximating functions in these Sobolev spaces.
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