Mathematical Analysis
The zero-one law for a complete orthonormal system
Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 335-337.

A complete orthonormal system of functions $Θ={θn}n=1∞,θn∈L[0,1]∞$ is constructed such that $∑n=1∞anθn$ converges almost everywhere on $[0,1]$ if ${an}n=1∞∈l2$ and $∑n=1∞anθn$ diverges a.e. for any ${an}n=1∞∉l2$. We also show that for any complete ONS ${fn}n=1∞$ of functions defined on $[0,1]$ there exists a fixed non decreasing subsequence ${nk}k=1∞$ of natural numbers such that for any $f∈L[0,1]0$ and some sequence of coefficients ${bn}n=1∞$,

 $∑n=1nkbnfn→fa.e. whenk→∞.$

On construit un système orthonormal complet $Θ={θn}n=1∞,θn∈L[0,1]∞$ tel que $∑n=1∞anθn$ converge presque partout pour n'importe quel ${an}n=1∞∈l2$ et diverge presque partout pour n'importe quel ${an}n=1∞∉l2$. Nous démontrons que pour toute système orthonormal complet ${fn}n=1∞$ il existe une sous suite croissante ${nk}k=1∞$ d'entiers naturels tels que pour tout $f∈L[0,1]0$ il existe une suite de coefficients tels que

 $∑n=1Nkbnfn→fp.p. sik→∞.$

Accepted:
Published online:
DOI: 10.1016/j.crma.2004.07.009
Kazarian, Kazaros 1

@article{CRMATH_2004__339_5_335_0,
author = {Kazarian, Kazaros},
title = {The zero-one law for a complete orthonormal system},
journal = {Comptes Rendus. Math\'ematique},
pages = {335--337},
publisher = {Elsevier},
volume = {339},
number = {5},
year = {2004},
doi = {10.1016/j.crma.2004.07.009},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2004.07.009/}
}
TY  - JOUR
AU  - Kazarian, Kazaros
TI  - The zero-one law for a complete orthonormal system
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 335
EP  - 337
VL  - 339
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2004.07.009/
DO  - 10.1016/j.crma.2004.07.009
LA  - en
ID  - CRMATH_2004__339_5_335_0
ER  - 
%0 Journal Article
%A Kazarian, Kazaros
%T The zero-one law for a complete orthonormal system
%J Comptes Rendus. Mathématique
%D 2004
%P 335-337
%V 339
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2004.07.009/
%R 10.1016/j.crma.2004.07.009
%G en
%F CRMATH_2004__339_5_335_0
Kazarian, Kazaros. The zero-one law for a complete orthonormal system. Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 335-337. doi : 10.1016/j.crma.2004.07.009. http://www.numdam.org/articles/10.1016/j.crma.2004.07.009/

[1] Arutyunyan, F.G. Representation of functions by multiple series, Akad. Nauk Armyan. SSR Dokl., Volume 64 (1977), pp. 72-76 (in Russian)

[2] Bourgain, J. On Kolmogorov's rearrangement problem for orthogonal systems and Garsia's conjecture, Lecture Notes in Math., vol. 1376, 1989, pp. 207-250

[3] Kashin, B.S. A certain complete orthonormal system, Mat. Sb., Volume 99 (141) (1976) no. 3, pp. 356-365 (in Russian) English translation Math. USSR-Sb., 28, 1976, pp. 315-324

[4] Kashin, B.S. On some properties of orthogonal systems of convergence, Trudy Mat. Inst. Steklov, Volume 143 (1977), pp. 68-87 (in Russian) English translation Proc. Steklov Inst. Math., 1, 1980, pp. 73-92

[5] Kashin, B.S.; Saakyan, A.A. Orthogonal Series, Transl. Math. Monographs, vol. 75, American Mathematical Society, Providence, RI, 1989

[6] Kazarian, K. A complete orthonormal system of divergence, C. R. Acad. Sci. Paris, Ser. I, Volume 337 (2003), pp. 85-88

[7] Kazarian, K. A complete orthonormal system of divergence, J. Funct. Anal., Volume 214 (2004), pp. 284-311

[8] Kazarian, K.S.; Kazarian, S.S. On the representations of functions of the $Lr,0⩽r<1$ spaces, Geometry, Analysis and Applications (Varanasi, 2000), World Sci. Publishing, River Edge, NJ, 2001, pp. 185-201

[9] Kazarian, K.S.; Waterman, D. Theorems on representations of functions by series, Mat. Sb., Volume 191 (2000) no. 12, pp. 123-140 (English translation Sb. Math., 191, 11–12, 2000, pp. 1873-1889)

[10] Marcinkiewicz, J. Sur la convergence des series orthogonales, Studia Math., Volume 67 (1936), pp. 39-45

[11] Menshov, D.E. Summation of the orthogonal series by linear methods, Izv. Akad. Nauk USSR Math. Ser. (1937), pp. 203-230 (in Russian)

[12] Pogosyan, N.B. Representation of measurable functions by bases in $Lp[0,1]$, $p⩾2$, Akad. Nauk Armyan. SSR Dokl., Volume 63 (1976), pp. 205-209 (in Russian)

[13] Ulyanov, P.L. Solved and unsolved problems in the theory of trigonometric and orthogonal series, Uspekhi Mat. Nauk, Volume 19 (1) (1964) no. 115, pp. 3-69 (in Russian) English translation Russian Math. Surveys, 19, 1964

[14] Talalyan, A.A. Representation of measurable functions by series, Uspekhi Mat. Nauk, Volume 15 (5) (1960) no. 95, pp. 77-141 (English translation Russian Math. Surveys, 15, 1960)

Cited by Sources: