Geodesic
Spherical partial sums of the double Fourier series of functions of bounded generalized variation
Sbornik. Mathematics, Tome 188 (1997) no. 1, pp. 29-60 
M. I. Dyachenko. Spherical partial sums of the double Fourier series of functions of bounded generalized variation. Sbornik. Mathematics, Tome 188 (1997) no. 1, pp. 29-60. http://geodesic.mathdoc.fr/item/SM_1997_188_1_a1/
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     title = {Spherical partial sums of the~double {Fourier} series of functions of bounded generalized variation},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The spherical partial sums of the double Fourier series of functions in the Waterman classes are studied. The main result of the paper is as follows. Theorem 1. {\it Let $\Lambda_\varepsilon =\biggl\{\dfrac{n^{3/4}}{(\ln(n+1))^{1/2+\varepsilon}}\biggr\}_{n=1}^\infty$ for $\varepsilon>0$. Let $f(x,y)\in\Lambda_\varepsilon BV(T^2)$ and let \begin{align*} I_r(f)&=\sup_{x,y\in T}\sup_{u,v\in[-1,1]}J_r(f) \\ &=\sup_{x,y\in T}\sup_{u,v\in[-1,1]}\sum_{r-1<|(m,n)|\leqslant r+1}|a_{m,n}(\psi_{x,y,u,v})|\leqslant C \end{align*} for $r\geqslant 1$, where $$ \psi _{x,y,u,v}(s,t)=\psi (s,t)=f(x+t,y+s)w(t)w(s)e^{-i(tu+sv)}, \quad and\quad w(\tau)=\frac\tau{2\sin(\theta/2)}\,. $$ Then $$ \sup_{R\geqslant 1}\sup _{(x,y)\in T^2}|S_R(f,x,y)|\leqslant C(f,\varepsilon). $$ for each $R\geqslant 1$.} Problem of circular convergence of Fourier series of the characteristic function of plane convex sets are also considered.

[1] Waterman D., “On convergence of Fourier series of functions of bounded generalized variation”, Studia Math., 44:1 (1972), 107–117 | MR | Zbl

[2] Chandrasekharan K., Minaksisundaram S., “Some results on double Fourier series”, Duke Math. J., 14:3 (1947), 731–753 | DOI | MR | Zbl

[3] Golubov B. I., “O skhodimosti sfericheskikh srednikh Rissa kratnykh ryadov i integralov Fure ot funktsii ogranichennoi obobschennoi variatsii”, Matem. sb., 89:4 (1972), 630–653 | MR | Zbl

[4] Dyachenko M. I., “Nekotorye problemy teorii kratnykh ryadov Fure”, UMN, 47:5 (1992), 97–162 | MR | Zbl

[5] Dyachenko M. I., “Waterman classes and spherical partial sums of double Fourier series”, Anal. Math., 21:1 (1995), 3–21 | DOI | MR | Zbl

[6] Yudin V. A., “Povedenie konstant Lebega”, Matem. zametki, 17:3 (1975), 401–405 | MR | Zbl

[7] Alimov Sh. A., Ilin V. A., “Usloviya skhodimosti spektralnykh razlozhenii, sootvetstvuyuschikh samosopryazhennym rasshireniyam ellipticheskikh operatorov, 1”, Differents. uravneniya, 7:4 (1971), 670–710 | MR | Zbl

[8] Dyachenko M. I., “Vypuklye mnozhestva i kratnye ryady Fure”, Tr. MIAN, 200, Nauka, M., 1991, 147–156 | MR | Zbl

[9] Stein E. M., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974 | Zbl

[10] Vatson G. N., Teoriya besselevykh funktsii, Ch. 1, IL, M., 1949

[11] Saakyan A. A., “O skhodimosti dvoinykh ryadov Fure funktsii ogranichennoi garmonicheskoi variatsii”, Izv. AN ArmSSR. Ser. matem., 21:6 (1986), 517–529 | MR

[12] Potapov M. K., “Izuchenie nekotorykh klassov funktsii pri pomoschi priblizheniya “uglom””, Tr. MIAN, 117, Nauka, M., 1972, 256–291 | MR | Zbl