# zbMATH — the first resource for mathematics

The Dirichlet ring and unconditional bases in $$L_2[0, 2\pi]$$. (English. Russian original) Zbl 1315.46015
Funct. Anal. Appl. 47, No. 3, 227-232 (2013); translation from Funkts. Anal. Prilozh. 47, No. 3, 75-81 (2013).
Summary: It is observed that the Dirichlet ring admits a representation in an infinite-dimensional matrix algebra. The resulting matrices are subsequently used in the construction of nonorthogonal Riesz bases in a separable Hilbert space. This framework enables custom design of a plethora of bases with interesting features. Remarkably, the representation of signals in any one of these bases is numerically implementable via fast algorithms.

##### MSC:
 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces 42C30 Completeness of sets of functions in nontrigonometric harmonic analysis 42C15 General harmonic expansions, frames
Full Text:
##### References:
 [1] T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York-Heidelberg-Berlin, 1976. · Zbl 0335.10001 [2] K. I. Babenko, ”On conjugate functions,” Dokl. Akad. Nauk SSSR, 62 (1948), 157–160. [3] A. Beurling, ”The collected works of Arno Beurling,” in: Harmonic Analysis, Contemp. Math., vol. 2, Birkhäuser, Boston, 1989, 378–380. [4] L. Carleson, ”On convergence and growth of partial sums of Fourier series,” Acta Math., 116 (1966), 135–157. · Zbl 0144.06402 · doi:10.1007/BF02392815 [5] K. Chandrasekharan, Arithmetical Functions, Springer-Verlag, New York-Heidelberg-Berlin, 1970. · Zbl 0217.31602 [6] P. Djakov and B. Mityagin, ”Bari-Markus property for Riesz projections of 1D periodic Dirac operators,” Math. Nachr., 283 (2010), 443–462. · Zbl 1198.34188 · doi:10.1002/mana.200910003 [7] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Math. Monographs, vol. 18, Amer. Math. Soc., Providence, RI, 1969. · Zbl 0181.13504 [8] H. Hedenmalm, P. Lindqvist, and K. Seip, ”A Hilbert space of Dirichlet series and systems of dilated functions in L 2(0, 1),” Duke Math. J., 86 (1997), 1–37. · Zbl 0887.46008 · doi:10.1215/S0012-7094-97-08601-4 [9] T. Kato, Perturbation Theory for Linear Operators; Corr. Printing of the 2nd Ed., Springer-Verlag, Berlin-Heidelberg-New York, 1980. · Zbl 0435.47001 [10] A. M. Olevskii, ”On operators generating conditional bases in a Hilbert space,” Mat. Zametki, 12:1 (1972), 73–84; English transl.: Math. Notes, 12:1 (1972), 476–482. [11] A. A. Shkalikov, ”On the basis problem of eigenfunctions of an ordinary differential operator,” Uspekhi Matem. Nauk, 34:5(209) (1979), 235–236; English transl.: Russian Math. Surveys, 34:5 (1979), 249–250. · Zbl 0471.34014 [12] A. Sowa, ”A fast-transform basis with hysteretic features,” in: IEEE Conference Proceedings: Electrical and Computer Engineering (CCECE), 2011 24th Canadian Conference on, 8–11 May 2011, 000253–000257. [13] A. Sowa, ”Factorizing matrices by Dirichlet multiplication,” Linear Algebra Appl., 438:5 (2013), 2385–2393. · Zbl 1263.15013 · doi:10.1016/j.laa.2012.09.021 [14] A. Sowa, ”On an eigenvalue problem with a reciprocal-linear term,” Waves in Random and Complex Media, 22:2 (2012), 186–206. · Zbl 1291.34141 · doi:10.1080/17455030.2011.636085 [15] R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York-London-Toronto, 1980. · Zbl 0493.42001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.