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• 1
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
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.
Type of Medium: Electronic Resource
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• 3
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
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
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
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.
Type of Medium: Electronic Resource
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• 7
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.
Type of Medium: Electronic Resource
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