Finite Parseval frames in Walsh analysis
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh Winter Mathematical School "Modern Methods of Function Theory and Related Problems." January 28 – February 2, 2019. Part 1, Tome 170 (2019), pp. 118-128.

Voir la notice de l'article provenant de la source Math-Net.Ru

Methods for constructing finite Parseval frames by using Walsh matrices and the discrete Vilenkin–Chrestenson transform are described.
Keywords: Walsh function, frame, orthogonal wavelet, Kadison–Singer problem.
Mots-clés : discrete transformation
@article{INTO_2019_170_a7,
     author = {Yu. A. Farkov},
     title = {Finite {Parseval} frames in {Walsh} analysis},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {118--128},
     publisher = {mathdoc},
     volume = {170},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2019_170_a7/}
}
TY  - JOUR
AU  - Yu. A. Farkov
TI  - Finite Parseval frames in Walsh analysis
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2019
SP  - 118
EP  - 128
VL  - 170
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2019_170_a7/
LA  - ru
ID  - INTO_2019_170_a7
ER  - 
%0 Journal Article
%A Yu. A. Farkov
%T Finite Parseval frames in Walsh analysis
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2019
%P 118-128
%V 170
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2019_170_a7/
%G ru
%F INTO_2019_170_a7
Yu. A. Farkov. Finite Parseval frames in Walsh analysis. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh Winter Mathematical School "Modern Methods of Function Theory and Related Problems." January 28 – February 2, 2019. Part 1, Tome 170 (2019), pp. 118-128. http://geodesic.mathdoc.fr/item/INTO_2019_170_a7/

[1] Agaev G. N., Vilenkin N. Ya., Dzhafarli G. M., Rubinshtein A. I., Multiplikativnye sistemy funktsii i garmonicheskii analiz na nulmernykh gruppakh, Elm, Baku, 1981 | MR

[2] Bespalov M. S., “Diskretnoe preobrazovanie Krestensona”, Probl. peredachi inform., 46:4 (2010), 91–115 | MR | Zbl

[3] Bespalov M. S., “Sobstvennye podprostranstva diskretnogo preobrazovaniya Uolsha”, Probl. peredachi inform., 46:3 (2010), 60–79 | MR | Zbl

[4] Golubov B. I., Efimov A. V., Skvortsov V. A., Ryady i preobrazovaniya Uolsha: Teoriya i primeneniya, Izd-vo LKI, M., 2008 | MR

[5] Dobeshi I., Desyat lektsii po veivletam, Regulyarnaya i khaoticheskaya dinamika, Izhevsk, 2001

[6] Zalmanzon L. A., Preobrazovaniya Fure, Uolsha, Khaara i ikh primenenie v upravlenii, svyazi i drugikh oblastyakh, Nauka, M., 1989

[7] Malla S., Veivlety v obrabotke signalov, Mir, M., 2005

[8] Malozemov V. N. Masharskii S. M., Osnovy diskretnogo garmonicheskogo analiza, Lan, SPb., 2012

[9] Novikov I. Ya., Protasov V. Yu., Skopina M. A., Teoriya vspleskov, Fizmatlit, M., 2006

[10] Robakidze M. G., Farkov Yu. A., “Primenenie funktsii Uolsha k postroeniyu freimov Parsevalya v prostranstvakh periodicheskikh posledovatelnostei”, Sovremennye problemy teorii funktsii i ikh prilozheniya, Mat. 19 Mezhdunar. Saratov. zimnei shkoly, posv. 90-letiyu akad. P. L. Ulyanova, Nauchnaya kniga, Saratov, 2018, 265–267

[11] Rodionov E. A., “O primenenii veivletov k tsifrovoi obrabotke signalov”, Izv. Saratov. un-ta. Nov. ser. Ser. Mat. Mekh. Inform., 16:2 (2016), 217–225 | MR | Zbl

[12] Ryabtsov I. S., “Neobkhodimye i dostatochnye usloviya prostoty freimov Parsevalya”, Vestn. SamGU. Estestvennonauch. ser., 2012, no. 6 (97), 42–48 | Zbl

[13] Farkov Yu. A., “Diskretnye veivlety i preobrazovanie Vilenkina—Krestensona”, Mat. zametki., 89:6 (2011), 914–928 | DOI | Zbl

[14] Farkov Yu. A., “Ortogonalnye vspleski v analize Uolsha”, Sovremennye problemy matematiki i mekhaniki, K 80-letiyu V. A. Skvortsova. Obobschennye integraly i garmonicheskii analiz, v. 11, Matematika, eds. Lukashenko T. P., Solodova A. P., Izd-vo Moskovskogo universiteta, M., 2016, 62–75

[15] Farkov Yu. A., “Parametricheskie mnozhestva dlya freimov v analize Uolsha”, Vestn. Evraz. nats. un-ta im. L. N. Gumileva. Ser. Mat. Inform. Mekh., 124:3 (2018), 89–94

[16] Albrecht A., Howlett P., Verma G., “Optimal splitting of Parseval frames using Walsh matrices”, Poincaré J. Anal. Appl., 2018, no. 2, 39–58 | MR | Zbl

[17] Balonin N. A., Doković D. Ž., Karbovskiy D. A., “Construction of symmetric Hadamard matrices of order $4\nu$ for $\nu=47,73,113$”, Spec. Matrices., 6 (2018), 11–22 | DOI | MR | Zbl

[18] Bownik M., “The Kadison–Singer problem”, Frames and Harmonic Analysis, Am. Math. Soc., Providence, Rhode Island, 2018, 63–92 | DOI | MR | Zbl

[19] Casazza P. G.,Tremain J. C., “The Kadison–Singer problem in mathematics and engineering”, Proc. Natl. Acad. Sci. USA., 103:7 (2006), 2032–2039 | DOI | MR | Zbl

[20] Christensen O., An Inroduction to Frames and Riesz Bases, Birkhäuser, Boston, 2016 | MR

[21] Farkov Yu. A., “Examples of frames on the Cantor dyadic group”, J. Math. Sci., 187:1 (2012), 22–34 | DOI | MR | Zbl

[22] Farkov Yu. A., “Constructions of MRA-based wavelets and frames in Walsh analysis”, Poincaré J. Anal. Appl., 2015, no. 2, 13–36 | MR | Zbl

[23] Farkov Yu. A., Lebedeva E. A., Skopina M. A., “Wavelet frames on Vilenkin groups and their approximation properties”, Int. J. Wavelets Multires. Inf. Process, 13:5 (2015), 1550036 | DOI | MR | Zbl

[24] Farkov Yu. A., “Wavelet frames related to Walsh functions”, Eur. J. Math., 5 (2019), 250–267 | MR | Zbl

[25] Farkov Yu. A., Manchanda P., Siddiqi A. H., Construction of Wavelets through Walsh Functions, Springer, Singapore, 2019 | MR | Zbl

[26] Hedayat A., Wallis W. D., “Hadamard matrices and their applications”, Ann. Stat., 6:6 (1978), 1184–1238 | DOI | MR | Zbl

[27] Kadison R. V., Singer I. M., “Extensions of pure states”, Am. J. Math., 81 (1959), 383–400 | DOI | MR | Zbl

[28] Krivoshein A., Protasov V., Skopina M., Multivariate Wavelet Frames, Springer, Singapore, 2016 | MR | Zbl

[29] Lang W. C., “Orthogonal wavelets on the Cantor dyadic group”, SIAM J. Math. Anal., 27:1 (1996), 305–312 | DOI | MR | Zbl

[30] Schipp F., Wade W. R., Simon P., Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger, New York, 1990 | MR | Zbl

[31] Seberry J., Wysocki B. J., Wysocki T. A., “On some applications of Hadamard matrices”, Metrika., 62:2 (2005), 221–239 | DOI | MR | Zbl

[32] Dyadic Walsh analysis from 1924 onwards Walsh–Gibbs–Butzer dyadic differentiation in science, eds. Stanković R. S., Butzer P. L., Schipp F., Wade W. R., Atlantis Press, Amsterdam, 2015 | MR

[33] Stevens M., The Kadison–Singer Property, Springer, Berlin, 2016 | MR | Zbl

[34] Waldron S., An Introduction to Finite Tight Frames, Birkhäuser, New York, 2018 | MR | Zbl

[35] Weaver N., “The Kadison–Singer problem in discrepancy theory”, Discrete Math., 278:1–3 (2004), 227–239 | DOI | MR | Zbl