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Transactions of the American Mathematical Society

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Mean convergence of orthogonal Fourier series of modified functions

Authors: Martin G. Grigorian, Kazaros S. Kazarian and Fernando Soria
Journal: Trans. Amer. Math. Soc. 352 (2000), 3777-3798
MSC (2000): Primary 42C15, 42C20
Published electronically: April 13, 2000
MathSciNet review: 1695022
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Abstract: We construct orthonormal systems (ONS) which are uniformly bounded, complete, and made up of continuous functions such that some continuous and even some arbitrarily smooth functions cannot be modified so that the Fourier series of the new function converges in the $L^{p} $-metric for any $p > 2. $ We prove also that if $\Phi $ is a uniformly bounded ONS which is complete in all the spaces $L _ {[0,1]} ^{p} , 1 \leq p < \infty $, then there exists a rearrangement $\sigma $ of the natural numbers $\mathbf{N} $such that the system $\Phi _{\sigma }= \{ \phi _{\sigma (n)} \}_{n=1}^{\infty }$ has the strong $L^{p}$-property for all $p>2$; that is, for every $2 \leq p < \infty $ and for every $ f \in L _ {[0,1]} ^{p} $ and $\epsilon > 0 $there exists a function $ f_ \epsilon \in L _ {[0,1]} ^{p} $ which coincides with $f$ except on a set of measure less than $\epsilon $ and whose Fourier series with respect to the system $\Phi _{\sigma }$ converges in $L _ {[0,1]} ^{p} . $

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  • [B] N. K. Bary, A treatise on trigonometric series. Vols. I, II, Authorized translation by Margaret F. Mullins. A Pergamon Press Book, The Macmillan Co., New York, 1964. MR 0171116
  • [GKK] J. García-Cuerva, K. S. Kazarian, and S. S. Kazarian, On Men′shov’s 𝐶-strong property, Acta Sci. Math. (Szeged) 58 (1993), no. 1-4, 253–260. MR 1264236
  • [G] Adriano M. Garsia, Topics in almost everywhere convergence, Lectures in Advanced Mathematics, vol. 4, Markham Publishing Co., Chicago, Ill., 1970. MR 0261253
  • [GW] Casper Goffman and Daniel Waterman, Some aspects of Fourier series, Amer. Math. Monthly 77 (1970), 119–133. MR 0252940,
  • [Gr] M. G. Grigoryan, Some properties of orthogonal systems, Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), no. 5, 75–105 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 43 (1994), no. 2, 261–289. MR 1252757,
  • [K1] Ivan Dimovski, Isomorphism of the quotient fields, generated by Bessel-type differential operators, Math. Nachr. 67 (1975), 101–107. MR 0380294,
  • [K2] Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0248482
  • [Ka] K. S. Kazaryan, Some questions of the theory of orthogonal series, Mat. Sb. (N.S.) 119(161) (1982), no. 2, 278–294, 304 (Russian). MR 675197
  • [KS] Kazaros Kazarian and Fernando Soria, On Fourier series of orthogonal systems for modified functions, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 1, 9–14 (English, with English and French summaries). MR 1320822
  • [Ko] Kozlov, V. Ya., On complete systems of orthogonal functions, Mat. Sb. 26(68) (1950), 351-364. (Russian) MR 12:174g
  • [L] Luzin, N. N., Integral and trigonometric series, 1916, Dissertation, Moscow; augmented reprint, GITTL, Moscow, 1951. (Russian). MR 14:2g
  • [M1] Menchoff, D. [Men'shov D.E.], Sur l'unicité du développement trigonométrique, C.R. Acad. Sci. Paris 163 (1916), 433-436.
  • [M2] Menchoff, D. [Men'shov D.E.], Sur les séries de Fourier des fonctions continues, Mat. Sb. 8(50) (1940), 493-518. MR 2:189g
  • [M3] Menchoff, D. [Men'shov D.E.], Sur la représentation des fonctions mesurables par des séries trigonométriques, Mat. Sb. 9(51) (1941), 667-692. MR 3:106a
  • [M4] Menchoff, D. [Men'shov D.E.], Sur les sommes partielles des séries trigonométriques, Mat. Sb. 20(62) (1947), 197-236. (Russian; French summary) MR 8:577a
  • [O] A. M. Olevskiĭ, Existence of functions with nonremovable Carleman singularities, Dokl. Akad. Nauk SSSR 238 (1978), no. 4, 796–799 (Russian). MR 0467138
  • [Or] Orlicz, W., Über die unabhängig von der Anordnung fast überall konvergenten Reihen, Bull. Acad. Polon. Sci. 81 (1927), 117-125.
  • [P] N. B. Pogosjan, Representation of measurable functions by orthogonal series, Mat. Sb. (N.S.) 98(140) (1975), no. 1(9), 102–112, 158–159 (Russian). MR 0487248
    N. B. Pogosjan, Representation of measurable functions by bases in 𝐿_{𝑝}[0,1], (𝑝≥2), Akad. Nauk Armjan. SSR Dokl. 63 (1976), no. 4, 205–209 (Russian, with Armenian summary). MR 0487249
  • [T] A. A. Talaljan, The representation of measurable functions by series, Russian Math. Surveys 15 (1960), no. 5, 75–136. MR 0125401,
  • [U] P. L. Ul′yanov, The works of N. N. Luzin in metric function theory, Uspekhi Mat. Nauk 40 (1985), no. 3(243), 15–70, 239 (Russian). MR 795185
  • [U1] P. L. Ul′janov, Solved and unsolved problems in the theory of trigonometric and orthogonal series, Uspehi Mat. Nauk 19 (1964), no. 1 (115), 3–69 (Russian). MR 0161085
  • [U2] P. L. Ul′yanov, Unconditional convergence and summability, Izv. Akad. Nauk SSSR. Ser. Mat. 22 (1958), 811–840 (Russian). MR 0103381
  • [Z] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776

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Additional Information

Martin G. Grigorian
Affiliation: Department of Radiophysics, Yerevan State University, Yerevan 375049, Armenia

Kazaros S. Kazarian
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Fernando Soria
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Keywords: Complete orthonormal system, C-strong property, modification of functions, rearrangements of systems, divergence in metric, universal series
Received by editor(s): November 26, 1997
Published electronically: April 13, 2000
Additional Notes: The research described in this publication was made possible in part by Grant PB97/0030 from DGES. The first two authors were also supported by Grant MVR000 from the I.S.F
Article copyright: © Copyright 2000 American Mathematical Society