Institute for applied mathematics

Prof. Dr. Karl Scherer

Publications

Articles in Journals and Proceedings

Trigonometric convolution operators with kernels having alternating signs and their degree of convergence. Jber.Deutsch.Math.-Verein. 70(1967),86-99 (with P.L.Butzer and R.J.Nessel).

Über die Fundamentalsätze der klassischen Approximationstheorie in abstrakten Räumen. Oberwolfach Conf.(1968) (P.L.Butzer-B. Sz.-Nagy, eds.) 113-125 (with P.L.Butzer)

On the fundamental theorems of D.Jackson, S.N.Bernstein and theorems of M. Zamansky and S.B. Steckin. Aequationes Math. 3(1969), 170-185 (with P.L.Butzer).

Dualität bei Interpolations- und Approximationsräumen. Dissertation, TH Aachen 1969, 76pp.

On the fundamental approximation theorems and their dual versions. J. Approx. Theory 3(1970), 87-100 (with P.L.Butzer).

On the best approximation of continous functions by splines. SIAM J. Numer. Anal. 7(1970),418-423.

Über die beste Approximation von Lp -Funktionen durch Splines. In: Proc. Internat.Conference Varna(1970)( B.Penkov-D.Vacov eds.), 277-286.

Approximation theorems for sequences of commutative operators in Banach spaces. Proc. Internat.Conference Varna(1970) (B.Penkov-D.Vacov eds.), 137-145 (with P.L.Butzer).

Jackson and Bernstein type inequalities for families of commutative operators in Banach spaces. J. Approx. Theory 5(1972), 308-342(with P.L.Butzer).

A remark on the characterization of Lip 1 by trigonometric best approximation. J. Approx. Theory 5(1972),213-215.

An equivalence theorem on best approximation of continous functions by algebraic polynomials. Applicable Anal. 1 (1972), 343-354 (with H.J.Wagner).

Über die Konvergenz von natürlichen interpolierenden Splines. In: Oberwolfach Conf.(1971) (P.L.Butzer-B. Sz.-Nagy, eds.), 487-492.

Zur besten approximation in Banachräumen mit Anwendungen auf ganze Funktionen. Forschungsberichte des Landes Nordrhein-Westfalen Nr. 2311 (1973), 51-74 (with J.Jungeburth-W.Trebels).

Über die Dualen von Banachräumen, die durch lineare Approximationsprozesse erzeugt werden, und Anwendungen für periodische Distributionen. Acta Math.Sci. Hungar. 23(1972), 343-365.

Eine Bemerkung zur Konvergenz Hermitescher Interpolationsprozesse 21(1973), 220-222 (with H.Esser).

On the Banach-Steinhaus theorem and approximation in locally convex spaces. Acta Sci. Math. (Szeged) 34(1973), 25-34 (with P.L.Butzer-U.Westphal).

On an extension of a theorem of S.N.Bernstein. Studia Sci. Math. Hung. 8(1973), 269-271.

Characterization of generalized Lipschitz classes by best approximation by splines. SIAM J. of Numer. Analysis 11(1974), 283-304.

A comparison approach to direct theorems for polynomial spline approximation. In: Proc. Internat. Conf. "Aproximation Theory ", Poznan 1972 Z.Ciesielskii-J.Musielak), 189-214.

Approximation theorems for polynomial spline operators. In: Proc. Internat. Conf. " Functionalanalysis and its Applications ", Madras 1973, Springer Lecture Notes 399. 438-467.

Stetigkeitsmoduli und beste Approximation durch polynominale Splines. In: "Splinefunktionen " (Tagungsband Oberwolfach 1973, BI-Verlag Zürich 1974), 289-302.

Best approximation by Chebychevian splines und generalized Lipschitz spaces (Proc. Colloq. "Konstruktive Theorie der Funktionen ", Cluj 1973).

Konvergenzordnungen bei Ein- und Mehrschrittverfahren für gewöhnliche Differentialgleichungen. Computing 12 (1974), 127-143 (with H. Esser).

Bernstein-type inequalities in a Banach space. Mat. Zametki 17 (1975), 925-937.

Direct and inverse theorems for best approximation by L-splines. In: " Spline Functions " (Proc. Internat. Symp., Karlsruhe 1975, Springer Lecture Notes 501), 119-131 (with H. Johnen).

On the equivalence of the K-functional and moduli of continuity and some applications. In:" Constr. Theory of functions of several variables " (Proc. Conf. Oberwolfach 1976, Springer Lecture Notes 571), 119-140 (with H. Johnen).

Some inverse theorems for best approximation by L-splines. In: " Approximation Theory II " (Proc.Internat. Symp. Austin 1976, G.G. Lorentz - C.K. Chui - L.L. Schumaker eds.), Acad. Press New York 1976), 549-555.

A constructive theory for approximation by splines with an arbitrary sequence of knot sets. In: " Approximation Theory " (Proc.Internat. Colloq. Bonn 1976, Springer Lecture Notes 556), 167-183 (with R. De Vore).

Optimal degree of approximation by splines. In: " Optimal Estimation in Approximation Theory " (Proc. Internat. Symp. Freudenstadt 1976, Plenum Press New York 1977), 139-149.

Interpolation of linear operators on Sobolev spaces. Annals of Mathematics 109 (1979), 583-599 (with R. De Vore).

Best approximation by piecewise polynomials with variable knots and degrees. J. Approximation Theory 26 (1979), 1-13 (with W. Dahmen).

Multi-dimensional spline approximation. SIAM J. Numer. Anal. 17 (1980), 380-402 (with W. Dahmen - R. De Vore).

A dual basis for L-splines and applications. J. Approximation Theory 29 (1980), 151-169 (with L.L. Schumaker).

Variable knot, variable degree spline approximation to xb. In: " Quantitative Approximation " (Proc. Internat. Symp. Bonn 1979, Acad. Press New York 1980), 121-131 (with R. De Vore).

An identity for the characteristic polynomials of the Lp- projections on splines for a geometric mesh. In " Approximation Theory III " (Proc.Internat. Conf. Austin 1980, E.W. Cheney ed., Acad. Press New York 1980), 517-522 (with K. Höllig).

On optimal global error bounds obtained by scaled local error bounds. Numer. Math. 36(1980/81), 151-176.

Direct and converse theorems for approximation of curves by polygons in Serdica 9 (1983), 407-413 (with G. Iliev).

Piecewise Pade - Approximation of e-x on [0,¥]. J.Approx. Theory 40 (1984), 84 - 100 (with H.U. Opitz).

A generalization of the Pade-Approximation to e-x on [0,¥]. In " Constructive Theory of functions ", Sofia 1984, 649-657 (with H.U. Opitz).

On the rational approximation of e-x on [0,¥]. Constructive Approximation 1 (1985),195-216 (with H.U. Opitz).

Cardinal Hermite Interpolation with box -splines I. Constr. Approx. 3(1987), 223-238 (with S. Riemenschneider).

Existence of best parametric interpolation by curves. SIAM J. Math. Anal. 20 (1989), 160-168 (with P.W. Smith).

Algorithms for computing best cubic parametric interpolation, in " Algorithms for Approximation II "(J.C. Mason - M.G. Cox eds.), Chapman and Hall 1990, 193-207 (with C. Rademacher).

Best interpolation with free nodes by closed curves. In: Math. Methods in CAGD (T. Lyche-L.L. Schumaker eds.) Academic Press, Boston 1989, 549-559.

Best parametric interpolation in Lp-norms. Monografias de la Academia de Ciencas de Zaragosa. No.2 (1990), 67-80 (with C. Rademacher).

Cardinal Hermite Interpolation with box-splines II. Numer. Math. 58 (1991), 133-149 (with S. Riemenschneider).

Extension of the problem of best interpolation parametric curves to L-splines. In " Curves and Surfaces "(P.J. Laurent- A. LeMehaute-L.L. Schumaker eds.). Academic Press 1991, 427-435.

The condition number of B-splines and related constants. In: "Open problems in Approximation Theory " (Voneshda Voda 1993) SCT Publishing, Singapore 1994, 180-191.

On best convex interpolation of curves. In:" Curves und Surfaces in Geometric Design " (P.J. Laurent-A.LeMehaute - L.L. Schumaker eds.). A.K. Peters, Wellesley 1994, 233-240 (with C. Henninger).

On local parametric approximation by polynomial curves. In: " Approximation Theory "( M.W.Müller- M.Felten-D.Mache eds.) Akademie-Verlag 1995, 285-292.

A general framework for high-accuracy parametric interpolation. Math.Comp. 66 (1997), 237-260 (with K. M orken).

On the sup-norm condition number of the multivariate triangular Bernstein Basis. In " Multivariate Approximation and Splines ", Mannheim 1996, ISNM 125, S.141-151 (with T.Lyche ).

New upper bounds for the B-spline basis condition number. East J.Approx. 2 (1996), p.331-342 (with A.Yu.Shadrin).

Uniqueness of best parametric interpolation by cubic spline curves. Constr. Approx. 13(1997), 393-419.

New upper bounds for the B-spline basis condition number: A proof of deBoor's 2k-conjecture. J.Approx.Theory 99(1999), 217-229 (with A.Yu.Shadrin).

Parametric polynomial curves of local approximation order 8. In: " Curves and Surface Fitting "(A.Cohen - C.Rabut - L.L.Schumaker eds.) Nashboro Press 2003,375-384.

On the p-norm Condition Number of the Multivariate Triangular Bernstein Basis. Journal Computational and Applied Mathematics 119(2000), 259-273 ( with T.Lyche).

On the L1-norm condition number of the univariate Bernstein basis. Constructive Aproximation 18(2002), 503-528 (with T.Lyche).

Lower bounds for Bernstein-Bezier condition number. In "Mathematical Methods for Curves and Surfaces " (T.Lyche- L.L. Schumaker eds.),
Nashville 2001, 433-443.

Weighted Norm equivalences and Preconditioning. In " Domain Decomposition Methods in Science and Engineering"(R. Kornhuber - R.Hoppe - J.Periaux - O.Pironneau - O.Widlund - J. Xu eds.), Berlin 2005, 405-414.

Best average interpolation with free nodes. In "Mathematical Methods for Curves and Surfaces " (M.Daehlen - K.Moerken - L.L. Schumaker eds.), Nashboro Press 2005.

Mixed Norm Condition Numbers for the univariate Bernstein Basis. In "Approximation and Probability ", Banach Center Publications, Volume 72(2006), 177-188.

Addendum to 'On the L1-norm condition number of the univariate Bernstein basis'. Constructive Approximation 24(2006), 239-243 (with T.Lyche).

Robust Norm Equivalencies and Optimal Preconditioners for Diffusion Problems. Math. Comp. 76(2007), 1141-1161 (with M. Griebel and M.A. Schweitzer).

Robust norm-equivalencies and preconditioning. In " Domain Decomposition Methods in Science and Engineering XVII"(U.Langer - M.Discacciati - D.Keyes - O.Widlund - W. Zulehner eds.), Berlin 2007, 373-380.

On the optimal condition numbers of the B-Spline basis. Manuscript, 17 pages. To appear in the proceedings of MAIA(2007).

A decomposition method for the Helmholtz problem. Manuscript, 12 pages, Bonn 2007 (with S. V. Nepomnyashihk).

Multilevel preconditioners for bilinear finite element approxmations of diffusion problems, Manuscript, 10 pages, Bonn 2008 (with S. V. Nepomnyashihk).
Habilitation " Spline Approximation and generalized Lipschitz spaces"

Books

  1. Approximationsprozesse und Interpolationsmethoden.
    BI-Hochschulskripten 826/826a , Mannheim 1968 (with P.L.Butzer).
  2. Approximation Theory.Proc. Internat. Colloquium, Bonn 1976 ( ed. with R. Schaback).Springer Lecture Notes 556, Springer 1976.
  3. Quantitative Approximation.Proc. Internat. Symposium, Bonn 1979 (ed. with R. DeVore). Acad. Press , NY 1980.

Unpublished manuscripts

  1. Approximation by piecewise polynomials with variable degrees and knots. SFB 72 - Preprint Nr.316, 1980.
  2. 0(h2n) approximation by piecewise polynomial curves of degree n in the plane . Manuscript 1995
  3. Weighted Jackson and Bernstein Inequalities. SFB 611 - Preprint Nr.120, 2002.
  4. Approximation Processes, Norm Equivalences and Applications to Preconditioning. SFB 611 - Preprint Nr., 2004.

List of research areas

  • Approximation Theory in Banach Spaces
  • Fourier Analysis and Approximation
  • Interpolation Theory of Banach Spaces
  • Rate of approximation by Splines (fixed knots)
  • Multivariate Splines
  • Approximation with variable degree and knots
  • Approximation by rational functions
  • Parametric Approximation
  • Rate of Approximation by parametric polynomial curves
  • Stability of the B-Spline- and Bernstein-Bezier Basis
  • Robust Preconditioning for Additive Multilevel Schemes

News

Prof. Dr. S. Müller erhält den diesjährigen Lehrpreis der Universität Bonn (07.07.17).

Prof. Dr. Michael Ortiz kommt als Bonn Research Chair ans IAM (Pressemitteilung, 29.7.2016).

Prof. Dr. S. Conti erhält den diesjährigen Lehrpreis der Universität Bonn (05.07.2016).

Prof. Dr. Karl-Theodor Sturm, Koordinator des Hausdorff Zentrums für Mathematik an der Universität Bonn, erhält für seine eigene Forschung einen begehrten Advanced Grant des Europäischen Forschungsrats (ERC). (Pressemitteilung 21.04.2016)

Prof. Dr. Stefan Müller has been invited to become full member of the scientific society “Academia Europaea” in November. Only European scientists, who have been recommended by a review board and are confirmed by a vote of the council, can join the society. (06.01.2016)

Contact

Managing Director: Prof. Dr. Anton Bovier
Chief Administrator: Dr. B. Doerffel
geschaeftsfuehrung@iam.uni-bonn.de
Imprint

Mailing address

Institute for Applied Mathematics
University of Bonn
Endenicher Allee 60
D-53115 Bonn / Germany