Geodesic
Generalized localization for the multiple Walsh–Fourier series of functions in $L_p$, $p\geqslant 1$
Sbornik. Mathematics, Tome 186 (1995) no. 2, pp. 181-196 
S. K. Bloshanskaya; I. L. Bloshanskii. Generalized localization for the multiple Walsh–Fourier series of functions in $L_p$, $p\geqslant 1$. Sbornik. Mathematics, Tome 186 (1995) no. 2, pp. 181-196. http://geodesic.mathdoc.fr/item/SM_1995_186_2_a1/
@article{SM_1995_186_2_a1,
     author = {S. K. Bloshanskaya and I. L. Bloshanskii},
     title = {Generalized localization for the~multiple {Walsh{\textendash}Fourier} series of functions in $L_p$, $p\geqslant 1$},
     journal = {Sbornik. Mathematics},
     pages = {181--196},
     year = {1995},
     volume = {186},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_186_2_a1/}
}
TY  - JOUR
AU  - S. K. Bloshanskaya
AU  - I. L. Bloshanskii
TI  - Generalized localization for the multiple Walsh–Fourier series of functions in $L_p$, $p\geqslant 1$
JO  - Sbornik. Mathematics
PY  - 1995
SP  - 181
EP  - 196
VL  - 186
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1995_186_2_a1/
LA  - en
ID  - SM_1995_186_2_a1
ER  - 
%0 Journal Article
%A S. K. Bloshanskaya
%A I. L. Bloshanskii
%T Generalized localization for the multiple Walsh–Fourier series of functions in $L_p$, $p\geqslant 1$
%J Sbornik. Mathematics
%D 1995
%P 181-196
%V 186
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1995_186_2_a1/
%G en
%F SM_1995_186_2_a1

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

In this paper the concept of generalized localization almost everywhere (GL) is analyzed for the multiple Fourier–Walsh series of functions in $L_p(T^N)$, $T^N=[0,1)^N$, $p\geqslant 1$ summable over rectangles. (For multiple trigonometric series and Fourier integrals GL was introduced and analyzed earlier by one of the authors.) If $p>1$, then it is proved that GL holds for double Walsh–Fourier series on any open set. It breaks down on any set $E\subset T^N$ which is not dense in $T^N$ if $N=2$ and $p=1$ and also in the class $\mathbb C$ if $N\geqslant 3$. All results on the Walsh system obtained in this paper are identical to the results on GL for Fourier series in the trigonometric system obtained earlier by one of the authors.

[1] Fine N. J., “On the Walsh functions”, Trans. Amer. Math. Soc., 65:3 (1949), 372–414 | DOI | MR | Zbl

[2] Schipp F., Wade W. R., Simon P., Pal J., Walsh series: An introduction to dyadic harmonic analysis, Akad. kiado, Budapest, 1990 | MR

[3] Golubov B. I., Efimov A. V., Skvortsov V. A., Ryady i preobrazovaniya Uolsha: Teoriya i primeneniya, Nauka, M., 1987 | MR | Zbl

[4] Tonelli L., Serie trigonometriche, Bologna, 1928

[5] Zigmund A., Trigonometricheskie ryady, T. 2, Mir, M., 1965 | MR

[6] Bloshanskii I. L., “Obobschennaya lokalizatsiya pochti vsyudu i skhodimost dvoinykh ryadov Fure”, DAN SSSR, 242:1 (1978), 11–13 | MR

[7] Bloshanskii I. L., “Ravnoskhodimost razlozhenii v kratnyi trigonometricheskii ryad Fure i integral Fure”, Matem. zametki, 18:2 (1975), 153–168 | MR

[8] Bloshanskii I. L., “O raskhodimosti ryada Fure pochti vsyudu na zadannom mnozhestve i skhodimosti k nulyu vne ego”, DAN SSSR, 280:4 (1985), 777–780 | MR

[9] Bloshanskii I. L., “Dva kriteriya slaboi obobschennoi lokalizatsii dlya kratnykh trigonometricheskikh ryadov Fure funktsii iz $L_p$, $p\geqslant 1$”, Izv. AN SSSR. Ser. matem., 49:2 (1985), 243–282 | MR

[10] Billard P., “Sur la convergence presque partout des series de Fourier–Walsh des fonctions de l'espace $L_2([\,0,1])$”, Studia Math., 28:3 (1966–1967), 363–388 | MR

[11] Hunt R., “Almost everywhere convergence of Walsh–Fourier series of $L_2$ functions”, Actes Congr. Int. Math., 2 (1970), 655–661 | MR

[12] Sjölin P., “An inequality of Paley and convergence a.e. of Walsh–Fourier series”, Arkiv för Math., 7:6 (1969), 551–570 | DOI | MR | Zbl

[13] Móricz F., “On the convergence of double orthogonal series”, Analysis Math., 2 (1976), 287–304 | DOI | MR | Zbl

[14] Bloshanskii I. L., “O kriteriyakh slaboi obobschennoi lokalizatsii v $N$-mernom prostranstve”, DAN SSSR, 271:6 (1983), 1294–1298 | MR

[15] Bloshanskii I. L., “Struktura i geometriya maksimalnykh mnozhestv skhodimosti i neogranichennoi raskhodimosti pochti vsyudu kratnykh ryadov Fure funktsii iz $L_1$, ravnykh nulyu na zadannom mnozhestve”, Izv. AN SSSR. Ser. matem., 53:4 (1989), 675–707

[16] Stein E. M., “On limits of sequences of operators”, Ann. Math., 74:1 (1961), 140–170 | DOI | MR | Zbl

[17] Getsadze R. D., “Nepreryvnaya funktsiya s raskhodyaschimsya pochti vsyudu kratnym ryadom Fure po sisteme Uolsha–Peli”, Matem. sb., 128:2 (1985), 269–286 | MR