## Functiones et Approximatio Commentarii Mathematici

### On maximal weight solutions in a truncated trigonometric matrix moment problem

Andreas Lasarow

#### Abstract

The truncated trigonometric matrix moment problem is a well studied object. In particular, one can find in the literature the extremal value concerning the weight assigned to some point of the unit circle within the solution set of that matrix moment problem for the so-called non-degenerate situation and some special measure which realizes the extremal value. The primary concern of the paper is to give a basic proof for the fact that this distinguished measure is uniquely determined by that extremal feature.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 43, Number 2 (2010), 117-128.

Dates
First available in Project Euclid: 9 December 2010

https://projecteuclid.org/euclid.facm/1291903393

Digital Object Identifier
doi:10.7169/facm/1291903393

Mathematical Reviews number (MathSciNet)
MR2767166

Zentralblatt MATH identifier
1217.30035

#### Citation

Lasarow, Andreas. On maximal weight solutions in a truncated trigonometric matrix moment problem. Funct. Approx. Comment. Math. 43 (2010), no. 2, 117--128. doi:10.7169/facm/1291903393. https://projecteuclid.org/euclid.facm/1291903393

#### References

• N.I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Oliver and Boyd, London (1965).
• D.Z. Arov, Regular $J$-inner matrix-functions and related continuation problems, Operator Theory: Adv. and Appl. 43, Birkhäuser, Basel (1990), 63--87.
• J.A. Ball, I. Gohberg, and L. Rodman, Interpolation of Rational Matrix Functions, Operator Theory: Adv. and Appl. 45, Birkhäuser, Basel (1990).
• A. Bultheel, P. González-Vera, E. Hendriksen, and O. Njåstad, Orthogonal rational functions and quadrature on the unit circle, Numer. Algorithms 3 (1992), 105--116.
• A. Bultheel, P. González-Vera, E. Hendriksen, and O. Njåstad, Orthogonal Rational Functions, Cambridge Monographs on Applied and Comput. Math. 5, Cambridge University Press, Cambridge (1999).
• M.J. Cantero, L. Moral, and L. Velázquez, Measures and para-orthogonal polynomials on the unit circle, East J. Approx. 8 (2002), 447--464.
• R. Cruz-Barroso and P. González-Vera, On reproducing kernels and para-orthogonal polynomials on the unit circle, Rev. Acad. Canaria Cienc. 15 (2003), 79--91.
• H. Dette and W.J. Studden, The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley & Sons, New York (1997).
• V.K. Dubovoj, B. Fritzsche, and B. Kirstein, Matricial Version of the Classical Schur Problem, Teubner-Texte zur Mathematik 129, Teubner, Leipzig (1992).
• A.J. Duran and P. Lopez-Rodriguez, Density questions for the truncated matrix moment problem, Canad. J. Math. 49 (1997), 708--721.
• C. Foiaş and A.E. Frazho, The Commutant Lifting Approach to Interpolation Problems, Operator Theory: Adv. and Appl. 44, Birkhäuser, Basel (1990).
• B. Fritzsche, B. Kirstein, and A. Lasarow, On rank invariance of moment matrices of nonnegative Hermitian-valued Borel measures on the unit circle, Math. Nachr. 263/264 (2004), 103--132.
• B. Fritzsche, B. Kirstein, and A. Lasarow, The matricial Carathéodory problem in both nondegenerate and degenerate cases, Operator Theory: Adv. and Appl. 165, Birkhäuser, Basel (2006), pp. 251--290.
• B. Fritzsche, B. Kirstein, and A. Lasarow, On a class of extremal solutions of the nondegenerate matricial Carathéodory problem, Analysis $($Munich$)$ 27 (2007), 109--164.
• Ja.L. Geronimus, Polynomials orthogonal on a circle and their applications (Russian), Zapiski Naučno-Issled. Inst. Mat. Meh. Har'kov. Mat. Obšč. 19 (1948), 35--120.
• W.B. Jones, O. Njåstad, and W.J. Thron, Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle, Bull. London Math. Soc. 21 (1989), 113--152.
• M.G. Kreĭ n, The ideas of P.L. Chebyshev and A.A. Markov in the theory of limit values of integrals and their further development (Russian), Usp. Mat. Nauk 5 (1951), 3--66.
• M.G. Kreĭ n and A.A. Nudelman, The Markov Moment Problem and Extremal Problems, AMS Translations 50, AMS, Providence, R. I. (1977).