What strong monotonicity condition on Fourier coefficients can make the ratio ‖𝑓-𝑆_{𝑛}(𝑓)‖/𝐸_{𝑛}(𝑓) be bounded?

Zhou, S. P.

doi:10.1090/S0002-9939-1994-1198461-3

What strong monotonicity condition on Fourier coefficients can make the ratio $\Vert f-S_ n(f)\Vert /E_ n(f)$ be bounded?
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Proc. Amer. Math. Soc. 121 (1994), 779-785 Request permission

Abstract:

Let $\{ {\phi _n}\} _{n = 1}^\infty$ be a positive increasing sequence and ${\phi _n}\hat f(n)$ decrease. We ask what exact conditions on $\{ {\phi _n}\}$ make $\left \| {f - {S_n}(f)} \right \|/{E_n}(f)$ bounded? The present paper will give a complete answer to it.

References

D. J. Newman and T. J. Rivlin, Approximation of monomials by lower degree polynomials, Aequationes Math. 14 (1976), no. 3, 451–455. MR 410181, DOI 10.1007/BF01835995

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Song Ping Zhou, A problem on approximation by Fourier sums with monotone coefficients, J. Approx. Theory 62 (1990), no. 2, 274–276. MR 1068436, DOI 10.1016/0021-9045(90)90040-W
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776