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Positive matrices in the Hardy space with prescribed boundary representations via the Kaczmarz algorithm

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Abstract

For a singular probability measure μ on the circle, we show the existence of positive matrices on the unit disc which admit a boundary representation on the unit circle with respect to μ. These positive matrices are constructed in several different ways using the Kaczmarz algorithm. Some of these positive matrices correspond to the projection of the Szegő kernel on the disc to certain subspaces of the Hardy space corresponding to the normalized Cauchy transform of μ. Other positive matrices are obtained which correspond to subspaces of the Hardy space after a renormalization, and so are not projections of the Szegő kernel. We show that these positive matrices are a generalization of a spectrum or Fourier frame for μ, and the existence of such a positive matrix does not require μ to be spectral.

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References

  1. D. Alpay, A. Dijksma and D. Volok, Schur multipliers and de Branges-Rovnyak spaces: the multiscale case, J. Operator Theory 61 (2009), 87–118.

    MathSciNet  MATH  Google Scholar 

  2. D. Alpay and H. Dym, Hilbert spaces of analytic functions, inverse scattering and operator models. I, Integral Equations Operator Theory 7 (1984), 589–641.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Alpay and H. Dym. Hilbert spaces of analytic functions, inverse scattering and operator models. II, Integral Equations Operator Theory 8 (1985). 145–180.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Alpay, S. Reich and D. Shoikhet, Rigidity theorems, boundary interpolation and reproducing kernels for generalized Schur functions, Comput. Methods Funct. Theory 9 (2009), 347–364.

    Article  MathSciNet  MATH  Google Scholar 

  5. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404.

    Article  MathSciNet  MATH  Google Scholar 

  6. D. N. Clark, One dimensional perturbations of restricted shifts, J. Anal. Math. 25 (1972), 169–191.

    Article  MathSciNet  MATH  Google Scholar 

  7. L. de Branges and J. Rovnyak, Square Summable Power Series, Holt, Rinehart and Winston, New York-Toronto, Ont.-London, 1966.

    MATH  Google Scholar 

  8. R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366.

    Article  MathSciNet  MATH  Google Scholar 

  9. D. E. Dutkay and P. E. T. Jorgensen, Affine fractals as boundaries and their harmonic analysis, Proc. Amer. Math. Soc. 139 (2011), 3291–3305.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Herr and E. Weber, Fourier series for singular measures, Axioms 6 (2017), no. 2:7, 13 pp.

    Article  Google Scholar 

  11. P. E. T. Jorgensen and S. Pedersen, Dense analytic subspaces in fractal L 2 -spaces, J. Anal. Math. 75 (1998), 185–228.

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Kaczmarz, Angenäherte auflösung von systemen linearer gleichungen, Bull. Int. Acad. Polon. Sci., Cl. Sci. Math. 35 (1937), 355–357.

    MATH  Google Scholar 

  13. S. Kwapień and J. Mycielski, On the Kaczmarz algorithm of approximation in infinite-dimensional spaces, Studia Math. 148 (2001), 75–86.

    Article  MathSciNet  MATH  Google Scholar 

  14. T. L. Lance and M. I. Stessin, Multiplication invariant subspaces of Hardy spaces, Canad. J. Math. 49 (1997), 100–118.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. G. Miamee and M. Pourahmadi, Wold decomposition, prediction and parameterization of stationary processes with infinite variance, Probab. Theory Related Fields 79 (1988), 145–164.

    Article  MathSciNet  MATH  Google Scholar 

  16. E. Nelson, Kernel functions and eigenfunction expansions, Duke Math. J. 25 (1957), 15–27.

    Article  MathSciNet  MATH  Google Scholar 

  17. A. G. Poltoratskiĭ, Boundary behavior of pseudocontinuable functions, Algebra i Analiz 5 (1993), 189–210.

    MathSciNet  Google Scholar 

  18. D. Sarason, Function theory and de Branges’s spaces, in Operator Theory: Operator Algebras and Applications, Part 1 (Durham, NH, 1988), American Mathematical Society, Providence, RI, 1990, pp. 495–502.

    Google Scholar 

  19. D. Sarason, Sub-Hardy Hilbert Spaces in the Unit Disk, John Wiley & Sons, Inc., New York, 1994.

    MATH  Google Scholar 

  20. M. I. Stessin, Wold decomposition of the Hardy space and Blaschke products similar to a contraction, Colloq. Math. 81 (1999), 271–284.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. S. Strichartz, Mock Fourier series and transforms associated with certain Cantor measures, J. Anal. Math. 81 (2000), 209–238.

    Article  MathSciNet  MATH  Google Scholar 

  22. R. S. Strichartz, Convergence of mock Fourier series, J. Anal. Math. 99 (2006), 333–353.

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Wold, A Study in the Analysis of Stationary Time Series, Almqvist and Wiksell, Stockholm, 1954.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Iowa State University, Ames, IA, 50011, USA

    John E. Herr & Eric S. Weber

  2. Department of Mathematics, University of Iowa, Iowa City, IA, 52242, USA

    Palle E. T. Jorgensen

Authors
  1. John E. Herr
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  2. Palle E. T. Jorgensen
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  3. Eric S. Weber
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Correspondence to Eric S. Weber.

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Herr, J.E., Jorgensen, P.E.T. & Weber, E.S. Positive matrices in the Hardy space with prescribed boundary representations via the Kaczmarz algorithm. JAMA 138, 209–234 (2019). https://doi.org/10.1007/s11854-019-0026-6

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  • DOI: https://doi.org/10.1007/s11854-019-0026-6

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