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Hankel matrix transforms and operators. (English) Zbl 1309.47028
The article under review concerns Hankel matrices and Hankel operators.
Reviewer’s remark: The paper contains errors. In particular, the paper contains the following result:
Theorem 3.1. A Hankel matrix is regular if and only if
(i) \(\lim_{n\to\infty}h_{n+k}=0\).
(ii) \(\lim_{n\to\infty}\sum_{k=1}^\infty h_{n+k}=1\).
(iii) \(\sup_n\sum_{k=1}^\infty|h_{n+k}|\leq M\).
Condition (i) is very strange. Clearly, \( \lim_{n\to\infty}h_{n+k}=\lim_{n\to\infty}h_n. \)
Condition (iii) does not make any sense. If the series converges, then \(\lim_{n\to\infty}\sum_{k=1}^\infty h_{n+k}=0\).
Condition (iii) is also strange. Clearly, the supremum in question involves a non increasing sequence.

MSC:
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
41A35 Approximation by operators (in particular, by integral operators)
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References:
[1] Horn R, Johnson C: Matrix Analysis. Cambridge University Press, Cambridge; 1985.
[2] Iohvidov I: Hankel and Toeplitz matrices and forms. Birkhäuser, Boston, Mass; 1982.
[3] Choi M: Tricks or treats with the Hilbert matrix. Am Math Monthly 1983, 90(5):301–312.
[4] Peller V: Hankel Operators and their Applications. Springer Monographs in Mathematics. Springer-Verlag, New York; 2003.
[5] Al-Homidan S: Hybrid methods for approximating Hankel matrix. Numer Algorithms 2003, 32(1):57–66.
[6] Al-Homidan S: Solving Hankel matrix approximation problem using semidefinite programming. J Comput Appl Math 2007, 202(2):304–314.
[7] Macinnes C: The solution to a structured matrix approximation problem using Grass-mann coordinates. SIAM J Matrix Anal Appl 1999, 21(2):446–453. (electronic)
[8] Suffridge T, Hayden T: Approximation by a Hermitian positive semidefinite Toeplitz matrix. SIAM J Matrix Anal Appl 1993, 14(3):721–734.
[9] Nehari Z: On bounded bilinear forms. Ann of Math 1957, 65(2):153–162.
[10] Beylkin G, Monzón L: On approximation of functions by exponential sums. Appl Comput Harmon Anal 2005, 19(1):17–48.
[11] Sharafeddin O, Bowen H, Kouri D, Hoffman D: Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J Comput Phys 1992, 100(2):294–296.
[12] Nachamkin J, Maggiore C: A Fourier Bessel transform method for efficiently calculating the magnetic field of solenoids. J Comput Phys 1980, 37(1):41–55.
[13] Higgins W, Munson D: A hankel transform approach to tomographic image reconstruction. IEEE Trans Med Im 1988, 7: 59–72.
[14] Maddox I: Elements of Functional Analysis. 2nd edition. Cambridge University Press, Cambridge; 1988.
[15] Vermes P: Infinite matrices summing every periodic sequence. Nederl Akad Wetensch Proc Ser A 58 = Indag Math 1955, 17: 627–633.
[16] Berg I, Wilansky A: Periodic, almost periodic, and semiperiodic sequences. Michigan Math J 1962, 9: 363–368.
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