# zbMATH — the first resource for mathematics

Traces of the discrete Hilbert transform with quadratic phase. (English. Russian original) Zbl 1293.42006
Proc. Steklov Inst. Math. 280, 248-262 (2013); translation from Tr. Mat. Inst. Steklova 280, 255-269 (2013).
In this paper the authors extend the study of the following complex-valued function $H(t,x)=(p.v.)\sum\limits_{n\in\mathbb Z\setminus\{0\}}(2\pi i n)^{-1}\exp(\pi i(tn^2+2xn)), \quad (t,x)\in \mathbb R^2.$ Further $$H|_x$$ denotes that $$x$$ is fixed while $$t$$ is variable, $$H|_t$$ is defined in a similar way. If $$t$$ is irrational and has the continued fraction representation with partial quotients $$\{k_j\}_{j\in\mathbb N}$$ and convergents $$\{a_j/q_j\}_{j\in\mathbb N}$$, then $$\Omega_t(\delta):=\min(\delta q^{1/2}_{j+1}, q_j^{-1/2})$$, $$\delta\in(q_{j+1}^{-1},q_j^{-1}]$$, $$j\in\mathbb N$$. By $$\overline{\omega}[f](\delta)$$ we denote the usual uniform modulus of continuity of a $$2$$- or $$1$$-periodic function. A function $$f$$ is of bounded weak $$p$$-variation on $$\mathbb I$$ if the approximation of this function in the uniform norm by step functions with $$N$$ steps is of order $$O(N^{-1/p})$$ as $$N\to\infty$$. The main result of the paper is \smallskip Theorem 1. Let $$t$$ and $$x$$ be irrational numbers. Then (1) $$\overline{\omega}[H|_t](\delta)\sim \Omega_t(\delta)$$, $$\quad \overline{\omega}[H|_x](\delta)\sim \Omega_x(\delta^{1/4})$$; (2) $$H|_t$$ and $$H|_x$$ are nonwhere differentiable functions; moreover, for any fixed $$y$$ and $$s$$ $\limsup\limits_{\delta\to 0}\frac{|H(t,y+\delta)-H(t,y)|}{\delta^{1/2}}\gg 1, \quad \limsup\limits_{\delta\to 0}\frac{|H(s+\delta,x)-H(s,x)|}{\delta^{\alpha(s)}}\gg 1,$ where $$\alpha(s)=1/4$$ if $$s$$ is irrational and $$\alpha(s)=1/2$$ if $$s$$ is rational; (3) uniformly in $$t$$, the restriction $$H|_t$$ is a function of bounded weak quadratic variation in the variable $$x$$ on $$[0,1)$$, and uniformly in $$x$$ the restriction $$H|_x$$ is a function of bounded weak quartic variation in the variable $$t$$ on $$[0,2)$$.

##### MSC:
 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
Full Text:
##### References:
 [1] G. I. Arkhipov and K. I. Oskolkov, ”On a Special Trigonometric Series and Its Applications,” Mat. Sb. 134(2), 147–157 (1987) [Math. USSR, Sb. 62 (1), 145–155 (1989)]. · Zbl 0665.42003 [2] G. H. Hardy and J. E. Littlewood, ”Some Problems of Diophantine Approximation. II: The Trigonometric Series Associated with the Elliptic v-Functions,” Acta Math. 37, 193–238 (1914). · JFM 45.0305.03 · doi:10.1007/BF02401834 [3] G. H. Hardy, Collected Papers of G.H. Hardy, Including Joint Papers with J.E. Littlewood and Others (Clarendon Press, Oxford, 1966), Vol. 1. [4] S. Jaffard, ”The Spectrum of Singularities of Riemann’s Function,” Rev. Mat. Iberoam. 12(2), 441–460 (1996). · Zbl 0889.26005 · doi:10.4171/RMI/203 [5] V. Jarník, ”Über die simultanen diophantischen Approximationen,” Math. Z. 33, 505–543 (1931). · JFM 57.1370.01 · doi:10.1007/BF01174368 [6] A. Ya. Khinchin, Continued Fractions (Fizmatgiz, Moscow, 1961; Univ. Chicago Press, Chicago, 1964). [7] K. I. Oskolkov, ”On Functional Properties of Incomplete Gaussian Sums,” Can. J. Math. 43, 182–212 (1991). · Zbl 0728.11039 · doi:10.4153/CJM-1991-010-0 [8] K. I. Oskolkov, ”A Class of I.M. Vinogradov’s Series and Its Applications in Harmonic Analysis,” in Progress in Approximation Theory: An International Perspective (Springer, New York, 1992), pp. 353–402. · Zbl 0815.42003 [9] K. I. Oskolkov, ”Schrödinger Equation and Oscillatory Hilbert Transforms of Second Degree,” J. Fourier Anal. Appl. 4(3), 341–356 (1998). · Zbl 0912.35052 · doi:10.1007/BF02476032 [10] K. I. Oskolkov, ”The Schrödinger Density and the Talbot Effect,” in Approximation and Probability (Inst. Math. Pol. Acad. Sci., Warsaw, 2006), Banach Center Publ. 72, pp. 189–219. · Zbl 1140.42300 [11] K. I. Oskolkov and M. A. Chakhkiev, ”On Riemann’s’ Nondifferentiable’ Function and Schrödinger Equation,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 269, 193–203 (2010) [Proc. Steklov Inst. Math. 269, 186–196 (2010)]. · Zbl 1207.26010 [12] I. M. Vinogradov, Foundations of Number Theory, 9th ed. (Nauka, Moscow, 1981) [in Russian]. · Zbl 0547.10001 [13] A. Zygmund, Trigonometric Series (Cambridge Univ. Press, Cambridge, 1959). · Zbl 0085.05601
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.