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Constructing polynomials of minimal growth

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Abstract

In [2] a cyclic diagonal operator on the space of functions analytic on the unit disk with eigenvalues (λ n ) is shown to admit spectral synthesis if and only if for each j there is a sequence of polynomials (p n ) such that lim n→∞ p n (λ k ) = δ j,k and lim sup n→∞ sup k>j |p n (λ k )|1/k ≤ 1. The author also shows, through contradiction, that certain classes of cyclic diagonal operators are synthetic. It is the intent of this paper to use the aforementioned equivalence to constructively produce examples of synthetic diagonal operators. In particular, this paper gives two different constructions for sequences of polynomials that satisfy the required properties for certain sequences to be the eigenvalues of a synthetic operator. Along the way we compare this to other results in the literature connecting polynomial behavior ([4] and [9]) and analytic continuation of Dirichlet series ([1]) to the spectral synthesis of diagonal operators.

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References

  1. 1.

    J. M. Anderson, D. Khavinson, and H. S. Shapiro, “Analytic continuation of Dirichlet series”, Revista Matematica Iberoamericana, 11(2), 453–476, 1995.

  2. 2.

    I. Deters, “A connection between operator topologies, polynomial interpolation, and synthesis of diagonal operators”, J.Math. Anal. Appl., 350, 354–359, 2009.

  3. 3.

    I. Deters and S. M. Seubert, “Cyclic vectors of diagonal operators on the space of functions analytic on a disk”, J.Math. Anal. Appl., 334, 1209–1219, 2007.

  4. 4.

    I. Deters and S. M. Seubert, “An application of entire function theory to the synthesis of diagonal operators on the space of entire functions”, Houston Journal Of Mathematics, 38, 201–207, 2012.

  5. 5.

    A. S. B. Holland, Introduction To The Theory Of Entire Functions (Academic Press, 1973).

  6. 6.

    G. Latta and G. Polya, Complex Variables (JohnWiley And Sons, Inc., 1973)

  7. 7.

    J. Marín Jr. and S. M. Seubert, “Cyclic vectors of diagonal operators on the space of entire functions”, J. Math. Anal. Appl., 320, 599–610, 2006.

  8. 8.

    W. Rudin, Real And Complex Analysis 3rd Edition (McGraw-Hill, 1987).

  9. 9.

    S. M. Seubert, “Spectral Synthesis Of Jordan Operators”, J.Math. Anal. Appl., 249, 652–667, 2000.

  10. 10.

    S. M. Seubert, “Spectral synthesis of diagonal operators on the space of entire functions”, Houston Journal of Mathematics, 34(3), 807–816, 2008.

  11. 11.

    S. M. Seubert and J. Gordon Wade, “Spectral synthesis of diagonal operators and representing systems on the space of entire functions”, J.Math. Anal. Appl., 344, 9–16, 2008.

  12. 12.

    R. V. Sibilev, “Uniqueness theorem for Wolff-Denjoy series”, St. PetersburgMath. J., 7, 145–68, 1996.

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Author information

Correspondence to Ian Deters.

Additional information

Original Russian Text © I. Deters, 2014, published in Izvestiya NAN Armenii. Matematika, 2014, No. 3, pp. 25–38.

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Cite this article

Deters, I. Constructing polynomials of minimal growth. J. Contemp. Mathemat. Anal. 49, 117–125 (2014). https://doi.org/10.3103/S1068362314030029

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MSC2010 numbers

  • 30B10
  • 30B50
  • 47B36
  • 47B38

Keywords

  • Polynomials construction
  • invariant subspaces
  • diagonal operators
  • spectral synthesis