| 1. |
- Aronsson, Gunnar
(author)
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Linear control theory applied to a minimum-maximum problem
- 1983
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In: Journal of Approximation Theory. - : Elsevier BV. - 0021-9045 .- 1096-0430. ; 38:1, s. 90-100
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Journal article (peer-reviewed)abstract
- A linearisation technique and well-established linear control theory are used to derive relevant information concerning the extremal functions in a minimum-maximum problem.
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| 2. |
- Kallioniemi, Henry
(author)
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The Landau problem on compact intervals and optimal numerical differentiation
- 1990
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In: Journal of Approximation Theory. - : Elsevier BV. - 0021-9045 .- 1096-0430. ; 63:1, s. 72-91
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Journal article (peer-reviewed)abstract
- Let m and n be given integers, 0 < m<n. Let f(x) be a real- or complex-valued function of a real variable x on an interval I such that f(n − 1)(x) is absolutely continuous and f(n)(x) is bounded.The Landau problem is estimating an intermediate derivative f(m)(x) when bounds for f(x) and f(n)(x) are given. In this paper we present uniform bounds for f(m)(x) in terms of uniform bounds of f(x) and f(n)(x). This improves earlier bounds given by H. Cartan by, roughly, a factor of 1(e4m).Our method is based on the approximation of f(m)(x) by the mth derivative of a polynomial interpolating f(x) at n points in I. A technique to study the sign variations of the Peano kernel earlier used by us, Schönhage, and Schneider is developed further. We also use results by Gusev and by Rivlin.Our method enables us to get estimates of the truncation error and of the magnification of errors in the values employed for f(x) in such approximations.
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| 3. |
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| 4. |
- Maligranda, Lech, et al.
(author)
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Notes on non-interpolation spaces
- 1989
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In: Journal of Approximation Theory. - : Elsevier BV. - 0021-9045 .- 1096-0430. ; 56:3, s. 333-347
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Journal article (peer-reviewed)
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| 5. |
- Alpan, Gökalp
(author)
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Extremal polynomials on a Jordan arc
- 2022
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In: Journal of Approximation Theory. - : Elsevier. - 0021-9045 .- 1096-0430. ; 276
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Journal article (peer-reviewed)abstract
- Let gamma be a C2+ Jordan arc and let gamma(0) be the open arc which consists of interior points of gamma. We find concrete upper and lower bounds for the limit of Widom factors for L-2(mu) extremal polynomials on gamma & nbsp;which was given in Widom (1969). In addition, we show that the upper bound for the limit supremum of Widom factors for the weighted Chebyshev polynomials which was obtained in Widom (1969) can be improved once two normal derivatives of the Green function do not agree at one point z is an element of gamma(0). We also show that if gamma(0) is not analytic then we have improved upper bounds.
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| 6. |
- Andersson, Johan
(author)
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Mergelyan's approximation theorem with nonvanishing polynomials and universality of zeta-functions
- 2013
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In: Journal of Approximation Theory. - : Elsevier. - 0021-9045 .- 1096-0430. ; 167, s. 201-210
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Journal article (peer-reviewed)abstract
- We prove a variant of the Mergelyan approximation theorem that allows us to approximate functions that are analytic and nonvanishing in the interior of a compact set K with connected complement, and whose interior is a Jordan domain, with nonvanishing polynomials. This result was proved earlier by the author in the case of a compact set K without interior points, and independently by Gauthier for this case and the case of strictly starlike compact sets. We apply this result on the Voronin universality theorem for compact sets K, where the usual condition that the function is nonvanishing on the boundary can be removed. We conjecture that this version of Mergelyan's theorem might be true for a general set K with connected complement and show that this conjecture is equivalent to a corresponding conjecture on Voronin Universality.
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| 7. |
- Aptekarev, Alexander, I, et al.
(author)
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Differential equations for the recurrence coefficients limits for multiple orthogonal polynomials from a Nevai class
- 2020
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In: Journal of Approximation Theory. - : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0021-9045 .- 1096-0430. ; 255
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Journal article (peer-reviewed)abstract
- A limiting property of the nearest-neighbor recurrence coefficients for multiple orthogonal polynomials from a Nevai class is investigated. Namely, assuming that the nearest-neighbor coefficients have a limit along rays of the lattice, we describe it in terms of the solution of a system of partial differential equations. In the case of two orthogonality measures the differential equations become ordinary. For Angelesco systems, the result is illustrated numerically.
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| 8. |
- Asekritova, Irina, et al.
(author)
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Interpolation of Multiparameter Approximation Spaces
- 2004
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In: Journal of Approximation Theory. - : Elsevier. - 0021-9045 .- 1096-0430. ; 129:2, s. 182-206
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Journal article (peer-reviewed)abstract
- We prove a general interpolation theorem for linear operators acting simultaneously in several approximation spaces which are defined by multiparametric approximation families. As a consequence, we obtain interpolation results for finite families of Besov spaces of various types including those determined by a given set of mixed differences.
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| 9. |
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| 10. |
- Asekritova, Irina, et al.
(author)
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Stability of Fredholm properties on interpolation Banach spaces
- 2020
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In: Journal of Approximation Theory. - : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0021-9045 .- 1096-0430. ; 260
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Journal article (peer-reviewed)abstract
- The main aim of this paper is to prove novel results on stability of the semi-Fredholm property of operators on interpolation spaces generated by interpolation functors. The methods are based on some general ideas we develop in the paper. This allows us to extend some previous work in literature to the abstract setting. We show an application to interpolation methods introduced by Cwikel-Kalton-Milman-Rochberg which includes, as special cases, the real and complex methods up to equivalence of norms and also some other well known methods of interpolation. A by-product of these results get the stability of isomorphisms on Calderon products of Banach function lattices. We also study the important characteristics in operator Banach space theory, the so-called modules of injection and surjection, and we prove interpolation estimates of these modules of operators on scales of the Calderon complex interpolation spaces. (C) 2020 Elsevier Inc. All rights reserved.
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