Yudin, V. A. Spherical sums of Fourier series in \(L_ p\). (English. Russian original) Zbl 0708.42008 Math. Notes 46, No. 2, 675-680 (1989); translation from Mat. Zametki 46, No. 2, 145-152 (1989). L\({}_ p\)-estimates for spherical sums of Fourier series on \(R^ 2\) are obtained. The order of growth for the logarithmical multiplier is found. The result isn’t correct for \(R^ m\), \(m\geq 3\). Reviewer: A.L.Brodskij Cited in 3 Documents MSC: 42B05 Fourier series and coefficients in several variables 42B15 Multipliers for harmonic analysis in several variables 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 42A24 Summability and absolute summability of Fourier and trigonometric series Keywords:L\({}_ p\)-estimates; spherical sums of Fourier series PDFBibTeX XMLCite \textit{V. A. Yudin}, Math. Notes 46, No. 2, 675--680 (1989; Zbl 0708.42008); translation from Mat. Zametki 46, No. 2, 145--152 (1989) Full Text: DOI References: [1] M. Riesz, ?Sur les fonctions conjugees,? Math. Z.,27, 218-244 (1927). · JFM 53.0259.02 [2] C. Fefferman, ?The multiplier problem for the ball,? Ann. Math.,77, 191-195 (1971). · Zbl 0212.09403 [3] V. A. Il’in, ?Problems of localization and convergence for Fourier series with respect to fundamental systems of functions of the Laplace operator,? Usp. Mat. Nauk,23, No. 2, 61-120 (1968). [4] K. I. Babenko, ?Convergence in the mean of multiple Fourier series and asymptotics of the Dirichlet kernel of spherical means,? Preprint No. 52, Inst. Prikl. Mat. (IPM), Akad. Nauk SSSR, Moscow (1971). [5] K. I. Babenko, ?Summability and convergence of expansions in eigenfunctions of a differential operator,? Mat. Sb.,91, No. 2, 147-201 (1973). [6] Sh. A. Alimov, V. A. Il’in, and E. M. Nikishin, ?Questions of convergence of multiple trigonometric series and spectral decompositions. I,? Usp. Mat. Nauk,31, No. 6, 28-83 (1976). [7] B. S. Mityagin and E. M. Nikishin, ?Divergence of spectral decompositions in the mean and almost everywhere,? Dokl. Akad. Nauk SSSR,212, No. 3, 551-522 (1973). · Zbl 0298.35045 [8] B. S. Mityagin, ?Multiplier-idempotents in symmetric function spaces,? Punkts, Anal. Prilozhen.,6, No. 3, 81-82 (1972). [9] A. Cordoba, ?The Kakeya maximal functions and the spherical summation multipliers,? Am. J. Math.,99, 1-22 (1977). · Zbl 0384.42008 [10] A. Cordoba, ?The multiplier problem for polygon,? Ann. Math.,105, 581-588 (1977). · Zbl 0361.42005 [11] A. Cordoba, ?A note on Bochner-Riesz operators,? Duke Math. J.,46, 505-511 (1979). · Zbl 0438.42013 [12] A. Cordqba, ?Geometric Fourier analysis,? Ann. Inst. Fourier (Grenoble),32, 215-226 (1982). [13] R. Cooke, ?A Cantor-Lebesgue theorem in two dimensions,? Proc. Am. Math. Soc.,30, 547-550 (1971). · Zbl 0222.42014 [14] E. Stein and G. Weiss, Introduction to Harmonic Analysis on Euclidean Spaces [Russian translation], Mir, Moscow (1974). [15] P. Sjölin, ?A note on Littlewood-Paley decompositions with arbitrary intervals,? J. Approx. Theory,48, 328-334 (1986). · Zbl 0606.42013 [16] J. Bourgain, ?On square functions on the trigonometric system,? Bull. Soc. Math. Belg.,37, 20-26 (1985). · Zbl 0613.42015 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.