# zbMATH — the first resource for mathematics

On cubic lacunary Fourier series. (English) Zbl 1031.42003
The author proves three interesting theorems. The introduction gives an interesting and detailed historical survey on the subject. The proofs are voluminous and need several new lemmas and known theorems. We recall the abstract which gives a good survey on the main results.
Author’s abstract: “For $$2<\beta<4$$, we analyze the behavior, near the rational points $$x= p\pi/q$$, of $$\sum^\infty_{n=1} n^{-\beta}\exp(ixn^3)$$, considered as a function of $$x$$. We expand this series into a constant term, a term on the order of $$(x- p\pi/q)^{(\beta- 1)/3}$$, a term linear in $$x-p\pi/q$$, a “chirp” term on the order of $$(x- p\pi/q)^{(2\beta- 1)/4}$$, and an error term on the order of $$(x- p\pi/q)^{\beta/2}$$. At every such rational point, the left and right derivatives are either both finite (and equal) or both infinite, in contrast with the quadratic series, where the derivative is often finite on one side and infinite on the other. However, in the cubic series, again in contrast with the quadratic case, the chirp term generally has a different set of frequencies and amplitudes on the right and left sides. Finally, we show that almost every irrational point can be closely approximated, in a suitable Diophantine sense, by rational points where the cubic series has an infinite derivative. This implies that when $$\beta\leq (\sqrt{97}- 1)/4= 2.212\dots$$, both the real and imaginary parts of the cubic series are differentiable almost nowhere”.

##### MSC:
 42A55 Lacunary series of trigonometric and other functions; Riesz products 26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
Full Text:
##### References:
 [1] Paul L. Butzer and Eberhard L. Stark, ”Riemann’s example” of a continuous nondifferentiable function in the light of two letters (1865) of Christoffel to Prym, Bull. Soc. Math. Belg. Sér. A 38 (1986), 45 – 73 (1987). · Zbl 0629.01013 [2] J. J. Duistermaat, Self-similarity of ”Riemann’s nondifferentiable function”, Nieuw Arch. Wisk. (4) 9 (1991), no. 3, 303 – 337. · Zbl 0760.26009 [3] P. Erdős, On the distribution of the convergents of almost all real numbers, J. Number Theory 2 (1970), 425 – 441. · Zbl 0205.34902 · doi:10.1016/0022-314X(70)90046-6 · doi.org [4] C.F. Gauss, Disquisitiones Arithmeticae, Lipsiae, 1801. [5] Joseph Gerver, The differentiability of the Riemann function at certain rational multiples of \?, Amer. J. Math. 92 (1970), 33 – 55. · Zbl 0203.05904 · doi:10.2307/2373496 · doi.org [6] Joseph Gerver, More on the differentiability of the Riemann function, Amer. J. Math. 93 (1971), 33 – 41. · Zbl 0228.26008 · doi:10.2307/2373445 · doi.org [7] G.H. Hardy, Weierstrass’s non-differentiable function, Trans. Amer. Math. Soc. 17 (1916) 301-325. · JFM 46.0401.03 [8] G.H. Hardy and J.E. Littlewood, A new solution of Waring’s problem, Quarterly J. Math. 48 (1920) 272-293. · JFM 47.0114.01 [9] D. R. Heath-Brown and S. J. Patterson, The distribution of Kummer sums at prime arguments, J. Reine Angew. Math. 310 (1979), 111 – 130. · Zbl 0412.10028 [10] M. Holschneider and Ph. Tchamitchian, Pointwise analysis of Riemann’s ”nondifferentiable” function, Invent. Math. 105 (1991), no. 1, 157 – 175. · Zbl 0741.26004 · doi:10.1007/BF01232261 · doi.org [11] Seiichi Itatsu, Differentiability of Riemann’s function, Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), no. 10, 492 – 495. · Zbl 0501.26004 [12] Stephane Jaffard, The spectrum of singularities of Riemann’s function, Rev. Mat. Iberoamericana 12 (1996), no. 2, 441 – 460. · Zbl 0889.26005 · doi:10.4171/RMI/203 · doi.org [13] Stéphane Jaffard and Yves Meyer, Wavelet methods for pointwise regularity and local oscillations of functions, Mem. Amer. Math. Soc. 123 (1996), no. 587, x+110. · Zbl 0873.42019 · doi:10.1090/memo/0587 · doi.org [14] Wolfram Luther, The differentiability of Fourier gap series and ”Riemann’s example” of a continuous, nondifferentiable function, J. Approx. Theory 48 (1986), no. 3, 303 – 321. · Zbl 0626.42008 · doi:10.1016/0021-9045(86)90053-5 · doi.org [15] Ernst Mohr, Wo ist die Riemannsche Funktion \sum _\?=1^\infty \?\?\? \?²\?/\?² nichtdifferenzierbar?, Ann. Mat. Pura Appl. (4) 123 (1980), 93 – 104 (German, with English summary). · Zbl 0456.26003 · doi:10.1007/BF01796541 · doi.org [16] Erwin Neuenschwander, Riemann’s example of a continuous, ’nondifferentiable’ function, Math. Intelligencer 1 (1978/79), no. 1, 40 – 44. , https://doi.org/10.1007/BF03023045 S. L. Segal, Riemann’s example of a continuous ’nondifferentiable’ function. II, Math. Intelligencer 1 (1978/79), no. 2, 81 – 82. · Zbl 0396.26003 · doi:10.1007/BF03023065 · doi.org [17] Hervé Queffelec, Dérivabilité de certaines sommes de séries de Fourier lacunaires, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A291 – A293 (French). · Zbl 0221.26008
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.