Cesàro transforms of Fourier coefficients of 𝐿^{{}∞}-functions

Xiao, Jie

doi:10.1090/S0002-9939-97-04040-9

Cesàro Transforms of Fourier Coefficients
of $L^{\infty }$ -functions

Author: Jie Xiao
Journal: Proc. Amer. Math. Soc. 125 (1997), 3613-3616
MSC (1991): Primary 26D15, 42A05, 42A16
DOI: https://doi.org/10.1090/S0002-9939-97-04040-9
MathSciNet review: 1415377
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Abstract: In this note, we show that Cesàro transforms of Fourier cosine or sine coefficients of any $L^{\infty }(0,\pi )$ -function are Fourier cosine or sine coefficients of some $BMO(0,\pi )$ -function.

1. E. Alšynbaeva, Transforms of Fourier coefficients of some classes of functions, Mat. Zametki 25 (1979), no. 5, 645–651, 798 (Russian). MR 539574
2. R. L. Anderson and E. Taflin, Explicit nonsoliton solutions of the Benjamin-Ono equation, Lett. Math. Phys. 7 (1983), no. 3, 243–248. MR 706214, https://doi.org/10.1007/BF00400440
3. G.H. Hardy, Notes on some points in the integral caculus LXVI, Messenger Math. 58 (1928), 50-52.
4. C.T. Loo, Note on the properties of Fourier coefficients, Amer. J. Math. 71 (1949), 269-282. MR 10:603e
5. David A. Stegenga, Bounded Toeplitz operators on 𝐻¹ and applications of the duality between 𝐻¹ and the functions of bounded mean oscillation, Amer. J. Math. 98 (1976), no. 3, 573–589. MR 0420326, https://doi.org/10.2307/2373807
6. Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
7. A. Zygmund, Trigonometric series: Vols. I, II, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968. MR 0236587