Functions with universal Fourier-Walsh series
Sbornik. Mathematics, Tome 211 (2020) no. 6, pp. 850-874 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove results on the existence of functions whose Fourier series in the Walsh system are universal in some sense or other in the function classes $L^p[0,1]$, $0, and $M[0,1]$. We also give a description of the structure of these functions. Bibliography: 30 titles.
Keywords: universal functions, Fourier-Walsh series, almost everywhere convergence.
Mots-clés : convergence
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M. G. Grigoryan. Functions with universal Fourier-Walsh series. Sbornik. Mathematics, Tome 211 (2020) no. 6, pp. 850-874. http://geodesic.mathdoc.fr/item/SM_2020_211_6_a3/

[1] G. D. Birkhoff, “Démonstration d'un théorème élémentaire sur les fonctions entières”, C. R. Acad. Sci. Paris, 189 (1929), 473–475 | Zbl

[2] J. Marcinkiewicz, “Sur les nombres dérivés”, Fundamenta Math., 24 (1935), 305–308 | DOI | Zbl

[3] V. G. Krotov, “On the smoothness of universal Marcinkiewicz functions and universal trigonometric series”, Soviet Math. (Iz. VUZ), 35:8 (1991), 24–28 | MR | Zbl

[4] I. Joó, “On the divergence of eigenfunction expansions”, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 32 (1989), 3–36 | MR | Zbl

[5] G. R. MacLane, “Sequences of derivatives and normal families”, J. Analyse Math., 2 (1952), 72–87 | DOI | MR | Zbl

[6] W. Luh, “Universal approximation properties of overconvergent power series on open sets”, Analysis, 6:2-3 (1986), 191–207 | DOI | MR | Zbl

[7] K.-G. Grosse-Erdmann, Holomorphe Monster und universelle Funktionen, Ph.D. thesis, Univ. of Trier, Trier, 1987, Mitt. Math. Sem. Giessen, 176, Selbstverlag des Math. Inst., Giessen, 1987, iv+84 pp. | MR

[8] D. E. Menshov, “O chastnykh summakh trigonometricheskikh ryadov”, Matem. sb., 20(62):2 (1947), 197–238 | MR | Zbl

[9] A. A. Talalyan, “O skhodimosti pochti vsyudu podposledovatelnostei chastnykh summ obschikh ortogonalnykh ryadov”, Izv. AN Arm. SSR. Ser. matem., 10:3 (1957), 17–34 | MR | Zbl

[10] P. L. Ul'yanov, “Representation of functions by series and classes $\varphi(L)$”, Russian Math. Surveys, 27:2 (1972), 1–54 | DOI | MR | Zbl

[11] V. G. Krotov, “Representation of measurable functions by series in the Faber–Schauder system, and universal series”, Math. USSR-Izv., 11:1 (1977), 205–218 | DOI | MR | Zbl

[12] A. M. Olevskii, “The existence of functions with unremovable Carleman singularities”, Soviet Math. Dokl., 19 (1978), 102–106 | MR | Zbl

[13] V. I. Ivanov, “Representation of functions by series in metric symmetric spaces without linear functionals”, Proc. Steklov Inst. Math., 189 (1990), 37–85 | MR | Zbl

[14] M. G. Grigorian, “On the representation of functions by orthogonal series in weighted $L^{p}$ spaces”, Studia Math., 134:3 (1999), 207–216 | DOI | MR | Zbl

[15] N. B. Pogosyan, “Predstavlenie izmerimykh funktsii bazisami $L_{p}[0, 1]$, ($p\geq 2$)”, Dokl. AN Arm. SSR, 63:4 (1976), 205–209 | MR | Zbl

[16] M. Zh. Grigoryan, “Predstavlenie funktsii klassov $L^{p}[0, 1]$, $1\leq p2$, ortogonalnymi ryadami”, Dokl. AN Arm. SSR, 67:5 (1978), 269–274 | Zbl

[17] M. G. Grigorian, “An example of universal orthogonal series”, J. Contemp. Math. Anal., 35:4 (2000), 23–43 | MR | Zbl

[18] G. G. Gevorkyan, K. A. Navasardyan, “On Walsh series with monotone coefficients”, Izv. Math., 63:1 (1999), 37–55 | DOI | DOI | MR | Zbl

[19] M. G. Grigoryan, “On the universal and strong $(L^1,L^\infty)$-property related to Fourier–Walsh series”, Banach J. Math. Anal., 11:3 (2017), 698–712 | DOI | MR | Zbl

[20] M. G. Grigoryan, A. A. Sargsyan, “On the universal function for the class $L^{p}[0,1]$, $p\in(0,1)$”, J. Funct. Anal., 270:8 (2016), 3111–3133 | DOI | MR | Zbl

[21] M. G. Grigoryan, L. N. Galoyan, “On the universal functions”, J. Approx. Theory, 225 (2018), 191–208 | DOI | MR | Zbl

[22] A. Sargsyan, M. Grigoryan, “Universal function for a weighted space $L_{\mu}^{1}[0,1]$”, Positivity, 21:3 (2017), 1457–1482 | DOI | MR | Zbl

[23] M. G. Grigoryan, A. A. Sargsyan, “The structure of universal functions for $L^p$-spaces, $p\in(0,1)$”, Sb. Math., 209:1 (2018), 35–55 | DOI | DOI | MR | Zbl

[24] M. G. Grigoryan, K. A. Navasardyan, “Universal functions in ‘correction’ problems guaranteeing the convergence of Fourier–Walsh series”, Izv. Math., 80:6 (2016), 1057–1083 | DOI | DOI | MR | Zbl

[25] A. Kolmogoroff, “Sur les fonctions harmoniques conjugées et les series de Fourier”, Fundamenta Math., 7 (1925), 23–28 | Zbl

[26] A. A. Talaljan, F. G. Arutjunjan, “On the convergence of Haar series to $+\infty$”, Amer. Math. Soc. Transl. Ser. 2, 72, Amer. Math. Soc., Providence, RI, 1968, 1–8 | DOI | MR | Zbl

[27] N. N. Luzin', “K' osnovnoi teoremѣ integralnago ischisleniya”, Matem. sb., 28:2 (1912), 266–294 | Zbl

[28] D. Menchoff, “Sur la convergence uniforme des séries de Fourier”, Matem. sb., 11(53):1-2 (1942), 67–96 | MR | Zbl

[29] J. L. Walsh, “A closed set of normal orthogonal functions”, Amer. J. Math., 45:1 (1923), 5–24 | DOI | MR | Zbl

[30] R. E. A. C. Paley, “A remarkable series of orthogonal functions. I”, Proc. London Math. Soc. (2), 34:4 (1932), 241–264 ; II, 265–279 | DOI | MR | Zbl | DOI | MR | Zbl