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• 1
Electronic Resource
Springer
Constructive approximation 11 (1995), S. 85-106
ISSN: 1432-0940
Keywords: Primary 41A20, 41A21 ; Secondary 41A60, 30C15 ; Padé approximation ; Incomplete rationals ; Incomplete polynomials ; Steepest descent ; Zeros ; Poles
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract We consider rational approximations of the form $$\left\{ {(1 + z)^{\alpha n + 1} \frac{{p_{cn} (z)}}{{q_n (z)}}} \right\}$$ in certain natural regions in the complex plane wherep cn andq n are polynomials of degreecn andn, respectively. In particular we construct natural maximal regions (as a function of α andc) where the collection of such rational functions is dense in the analytical functions. So from this point of view we have rather complete analog theorems to the results concerning incomplete polynomials on an interval. The analysis depends on an examination of the zeros and poles of the Padé approximants to (1+z)αn+1. This is effected by an asymptotic analysis of certain integrals. In this sense it mirrors the well-known results of Saff and Varga on the zeros and poles of the Padé approximant to exp. Results that, in large measure, we recover as a limiting case. In order to make the asymptotic analysis as painless as possible we prove a fairly general result on the behavior, inn, of integrals of the form $$\int_0^1 {[t(1 - t)f_z (t)]^n {\text{ }}dt,}$$ wheref z (t) is analytic inz and a polynomial int. From this we can and do analyze automatically (by computer) the limit curves and regions that we need.
Type of Medium: Electronic Resource
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• 2
Electronic Resource
Springer
Constructive approximation 11 (1995), S. 439-453
ISSN: 1432-0940
Keywords: 41A15 ; 41A36 ; 41A63 ; Bernstein-Schoenberg operator ; simplex spline ; asymptotic error
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract We study an approximation operator of Bernstein-Schoenberg type which employs multivariate B-splines and was introduced in [2]. By first considering its action on convex functions, we derive a general formula of Voronovskaya type for the asymptotic error as the degree tends to infinity.
Type of Medium: Electronic Resource
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• 3
Electronic Resource
Springer
Constructive approximation 11 (1995), S. 423-438
ISSN: 1432-0940
Keywords: 41A63 ; 41A25 ; 42B ; 65D10 ; Approximation order ; Fourier transform ; Quasi-interpolation ; Cardinal interpolation ; Shift-invariant space
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract A Fourier analysis approach is taken to investigate the approximation order of scaled versions of certain linear operators into shift-invariant subspaces ofL 2(R d ). Quasi-interpolants and cardinal interpolants are special operators of this type, and we give a complete characterization of the order in terms of some type of ellipticity condition for a related function. We apply these results by showing that theL 2-approximation order of a closed shift-invariant subspace can often be realized by such an operator.
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• 4
Electronic Resource
Springer
Constructive approximation 11 (1995), S. 455-476
ISSN: 1432-0940
Keywords: 41A21 ; 65F05 ; (matrix) Padé approximation ; (vector) Padé-Hermite approximation ; (block) Hankel systems of equations ; look-ahead ; perfect points
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract In this paper, we present an algorithm to compute vector Padé-Hermite approximants along a sequence of perfect points in the vector Padé-Hermite table. We show the connection to matrix Padé approximants. The algorithm is used to compute the solution of a block Hankel system of linear equations.
Type of Medium: Electronic Resource
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• 5
Electronic Resource
Springer
Constructive approximation 11 (1995), S. 513-515
ISSN: 1432-0940
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Type of Medium: Electronic Resource
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• 6
Electronic Resource
Springer
Constructive approximation 11 (1995), S. 477-512
ISSN: 1432-0940
Keywords: 42C05 ; 33D15 ; Orthogonal polynomials ; Orthogonal Laurent polynomials ; Recurrence relation ; Hahn-Extonq-Bessel function ; q-Lommel polynomials ; Al-Salam-Chihara polynomials ; Continuousq-Hermite polynomials ; Chebyshev polynomials ; Perturbation
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Laurent polynomials related to the Hahn-Extonq-Bessel function, which areq-analogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurentq-Lommel polynomials are orthogonal is given. The strong moment functional gives rise to two positive definite moment functionals. For the corresponding sets of orthogonal polynomials, the orthogonality measure is determined using the three-term recurrence relation as a starting point. The relation between Chebyshev polynomials of the second kind and the Laurentq-Lommel polynomials and related functions is used to obtain estimates for the latter.
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• 7
Electronic Resource
Springer
Constructive approximation 12 (1996), S. 1-30
ISSN: 1432-0940
Keywords: Primary 28A75, 42B10, 46L55 ; Secondary 05B45 ; Iterated function system ; Affine maps ; Fractional measure ; Harmonic analysis ; Hilbert space ; Operator algebras
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract We consider affine systems inR n constructed from a given integral invertible and expansive matrixR, and a finite setB of translates,σ bx:=R–1x+b; the corresponding measure μ onR n is a probability measure fixed by the self-similarity $$\mu = \left| B \right|^{ - 1} \sum\nolimits_{b \in B} {\mu o\sigma _b^{ - 1} }$$ . There are twoa priori candidates for an associated orthogonal harmonic analysis: (i) the existence of some subset Λ inR n such that the exponentials {eiλ·x}Λ form anorthogonal basis forL 2(μ); and (ii) the existence of a certaindual pair of representations of theC *-algebraO N wheren is the cardinality of the setB. (For eachN, theC *-algebraO N is known to be simple; it is also called the Cuntz algebra.) We show that, in the “typical” fractal case, the naive version (i) must be rejected; typically the orthogonal exponentials inL 2(μ) fail to span a dense subspace. Instead we show that theC *-algebraic version of an orthogonal harmonic analysis, namely (ii), is a natural substitute. It turns out that this version is still based on exponentialse iλ·x, but in a more indirect way. (See details in Section 5 below.) Our main result concerns the intrinsic geometric features of affine systems, based onR andB, such that μ has theC *-algebra property (ii). Specifically, we show that μ has an orthogonal harmonic analysis (in the sense (ii)) if the system (R, B) satisfies some specific symmetry conditions (which are geometric in nature). Our conditions for (ii) are stated in terms of two pieces of data: (a) aunitary generalized Hadamard matrix, and (b) a certainsystem of lattices which must exist and, at the same time, be compatible with the Hadamard matrix. A partial converse to this result is also given. Several examples are calculated, and a new maximality condition for exponentials is identified.
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• 8
Electronic Resource
Springer
Constructive approximation 12 (1996), S. 67-94
ISSN: 1432-0940
Keywords: 41A10 ; 41A25 ; 41A28 ; Simultaneous approximation ; Timan-Gopengauz type estimates ; Ditzian-Totik moduli of smoothness
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Some estimates for simultaneous polynomial approximation of a function and its derivatives are obtained. These estimates are exact in a certain sense. In particular, the following result is derived as a corollary: Forf∈C r[−1,1],m∈N, and anyn≥max{m+r−1, 2r+1}, an algebraic polynomialP n of degree ≤n exists that satisfies $$\left| {f^{\left( k \right)} \left( x \right) - P_n^{\left( k \right)} \left( {f,x} \right)} \right| \leqslant C\left( {r,m} \right)\Gamma _{nrmk} \left( x \right)^{r - k} \omega ^m \left( {f^{\left( r \right)} ,\Gamma _{nrmk} \left( x \right)} \right),$$ for 0≤k≤r andx ∈ [−1,1], where ωυ(f(k),δ) denotes the usual vth modulus of smoothness off (k), and Moreover, for no 0≤k≤r can (1−x 2)( r−k+1)/(r−k+m)(1/n2)(m−1)/(r−k+m) be replaced by (1-x2)αkn2αk-2, with αk〉(r-k+a)/(r-k+m).
Type of Medium: Electronic Resource
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• 9
Electronic Resource
Springer
Constructive approximation 12 (1996), S. 95-110
ISSN: 1432-0940
Keywords: 41A05 ; 41A10 ; 65D05 ; Wavelets ; Chebyshev polynomials ; Interpolation ; Reconstruction ; Decomposition
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract We investigate a polynomial wavelet decomposition of theL 2(−1, 1)-space with Chebyshev weight, where the wavelets fulfill certain interpolatory conditions. For this approach we obtain the two-scale relations and decomposition formulas. Dual functions and Riesz-stability are discussed.
Type of Medium: Electronic Resource
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• 10
Electronic Resource
Springer
Constructive approximation 12 (1996), S. 31-65
ISSN: 1432-0940
Keywords: Primary 41A46, 68U10 ; Secondary 41A99, 54D35 ; Hausdorff distance ; Hausdorff continuity ; Completed graph ; Pixel function ; Image compression ; Fractal transform operator
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract The problem of finding appropriate mathematical objects to model images is considered. Using the notion of acompleted graph of a bounded function, which is a closed and bounded point set in the three-dimensional Euclidean spaceR 3, and exploring theHausdorff distance between these point sets, a metric spaceIM D of functions is defined. The main purpose is to show that the functionsf∈IM D, defined on the squareD=[0,1]2, are appropriate mathematical models of real world images. The properties of the metric spaceIM D are studied and methods of approximation for the purpose of image compression are presented. The metric spaceIM D contains the so-calledpixel functions which are produced through digitizing images. It is proved that every functionf∈IM D may be digitized and represented by a pixel functionp n, withn pixels, in such a way that the distance betweenf andp n is no greater than 2n −1/2. It is advocated that the Hausdorff distance is the most natural one to measure the difference between two pixel representations of a given image. This gives a natural mathematical measure of the quality of the compression produced through different methods.
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