Optimal Approximation by Blaschke Forms and Rational Functions ∗ Tao Qian † and Elias Wegert ‡ November 16, 2011 Abstract We study best approximation to functions in Hardy H 2 (D) by two classes of functions of which one is n-partial fractions with poles outside the closed unit disc and the other is n-Blaschke forms. Through the equal relationship between the two classes we obtain the existence of the minimizers in both classes. The algorithm for the minimizers for small orders are practical. Key Words: Approximation by rational functions, rational orthogonal system, Takenaka- Malmquist system, analytic signal, instantaneous frequency, adaptive decomposition, mono-components 1 Introduction By modified Blaschke products (of finite order ) we mean the functions B 1 (z ) = 1 √ 2π  1 −|a 1 | 2 1 − a 1 z , B 2 (z )= 1 √ 2π  1 −|a 2 | 2 1 − a 2 z z − a 1 1 − a 1 z , ..., B k (z )= 1 √ 2π  1 −|a k | 2 1 − a k z k−1  j =1 z − a j 1 − a j z , ..., (1.1) where the sequence {a k } ∞ k=1 defining the system is contained in D, where D stands for the open unit disc. {B k } ∞ k=1 is an orthonormal system, regarded as rational orthogonal (or Takenaka-Malmquist ) system ([1] and its references). We also use the notation B k = B {a 1 ,...,a k } to indicate the dependence of B k on the k-tuple {a 1 , ..., a k }. The system has been well studied since 1920’s with ample applications in the applied mathematics, including control theory, system identification, and signal analysis. If, in particular, a k = 0 for all k, then the system reduces to the Fourier basis { 1 √ 2π , z √ 2π , z 2 √ 2π , ..., z n √ 2π , ...}. * The work was supported by Macao FDCT 014/2008/A1, Research Grant of the University of Macau No. RG- UL/07-08s/Y1/QT/FSTR and DAAD A/09/00721 † Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Macao, China. E-mail:fsttq@umac.mo ‡ TU Bergakademie Freiberg, Germany. E-mail:wegert@math.tu-freiberg.de 1