Constructive rational approximation and interpolation in the unit disk

April 25, 2014

Christer Glader My research interests are in the area of approximation theory in the complex unit disk. I have studied finite Blaschke products in connection with constructive methods for rational Chebyshev approximation, (7), (9), and various types of rational interpolation problems, so-called Nevanlinna-Pick interpolation, in the unit disk, (5), (6). Applications of such methods can be found in H∞-optimization and control theory (model reduction, model matching theory etc.).

I have also considered rational unimodular boundary interpolation on the complex unit circle (interpolation with ratios of finite Blaschke products), (2), (4), and nonlinear Riemann-Hilbert problems with circular target curves (problems of finding all functions in the disk algebra satisfying a given boundary condition on the unit circle), (1), (3). In the rational cases these problems involve application of Wiener-Hopf factorization and generalized Nevanlinna-Pick interpolation by finite Blaschke products, and they have applications for instance in H∞-optimization of dynamical systems.

Some references

(1) C. Glader, E. Wegert: Nonlinear Rational Riemann-Hilbert Problems with Circular Target Curves, Computational Methods and Function Theory, Vol. 9 (2009), No. 2, 653-678.

(2) C. Glader: Minimal Degree Rational Unimodular Interpolation on the Unit Circle, Electronic Transactions on Numerical Analysis, Vol. 30(2008), 88-106.

(3) C. Glader, E. Wegert:Nonlinear Riemann-Hilbert Problems with Circular Target Curves, Mathematische Nachrichten, Vol. 281 (2008), No. 9, 1221-1239.

(4) C. Glader: Rational Unimodular Interpolation on the Unit Circle, Computational Methods and Function Theory, Vol. 6 (2006), No. 2, 481-492.

(5) C. Glader, M. Lindström: A new algorithm for meromorphic Nevanlinna-Pick interpolation, Numerische Mathematik, Vol. 100 (2005), No. 1, 49-69.

(6) C. Glader, M. Lindström: Finite Blaschke product interpolation on the closed unit disc, Journal of Mathematical Analysis and Applications, Vol. 273 (2002), No. 2, 417-427.

(7) C. Glader: A method for rational Chebyshev approximation of rational functions on the unit disk and on the unit interval, Numerical Algorithms, Vol. 26 (2001), 151-165.

(8) C. Glader: An Exchange Algorithm for Laurent Polynomial Chebyshev Approximation on the Unit Circle, Complex Variables, Vol. 46(2001), No. 4, 369-384

(9) C. Glader, G. Högnäs: An Equioscillation Characterization of Finite Blaschke Products, Complex Variables, Vol. 42 (2000), 107-118.