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[ TM1WLUESPLI ] UE Splines
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Workload Education level Study areas Responsible person Hours per week Coordinating university
1,5 ECTS B3 - Bachelor's programme 3. year Mathematics Markus Passenbrunner 1 hpw Johannes Kepler University Linz
Detailed information
Original study plan Bachelor's programme Technical Mathematics 2024W
Objectives Practice and Consolidation of important topics in the theory of splines
Subject Kapitel 1. Introduction

Kapitel 2. Prologue

  1. Best Approximation
  2. Interpolation of functions
  3. Divided Differences
  4. Total Positivity of Matrices

Kapitel 3. Polynomial functions

  1. Defintions
  2. Inequalities of Bernstein, Szegö and Markov
  3. Lp -Norms of polynomials
  4. Degree of Approximation of Polynomials

Kapitel 4. 1D Spline functions

  1. piecewise linear functions
  2. piecewise polynomials
  3. B-Splines
  4. Dual Funktionals to B-Splines
  5. Degree of Approximation of Spline functions
  6. Refining knot sequences
  7. Collocation

Kapitel 5. Higher dimenional spline functions

  1. Definitions and simple properties
  2. Recursion formulae
  3. Examples
Criteria for evaluation “Tick exercise” + Blackboard performance
Language English and French
Study material [1] H. B. Curry and I. J. Schoenberg. On Pólya frequency functions. IV. The fundamental spline functions and their limits. J. Analyse Math., 17:71–107, 1966.

[2] C. de Boor. Splinefunktionen. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1990.

[3] S. Demko. Inverses of band matrices and local convergence of spline projections. SIAM J. Numer. Anal., 14(4):616–619, 1977.

[4] R. A. DeVore and G. G. Lorentz. Constructive approximation, volume 303 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin,1993.

[5] L. L. Schumaker. Spline functions: basic theory. Cambridge Mathematical Library. Cambridge University

[6] A. Y. Shadrin. The L∞ -norm of the L2 -spline projector is bounded independently of the knot sequence: a proof of de Boor’s conjecture. Acta Math., 187(1):59–137, 2001.
Changing subject? No
On-site course
Maximum number of participants 25
Assignment procedure Direct assignment