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
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Proc. Amer. Math. Soc. 125 (1997), 3613-3616 Request permission

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.

References

E. Alšynbaeva, Transforms of Fourier coefficients of some classes of functions, Mat. Zametki 25 (1979), no. 5, 645–651, 798 (Russian). MR 539574
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, DOI 10.1007/BF00400440
G.H. Hardy, Notes on some points in the integral caculus LXVI, Messenger Math. 58 (1928), 50-52.
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
David A. Stegenga, Bounded Toeplitz operators on $H^{1}$ and applications of the duality between $H^{1}$ and the functions of bounded mean oscillation, Amer. J. Math. 98 (1976), no. 3, 573–589. MR 420326, DOI 10.2307/2373807
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
A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587