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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 \([0,2)\).

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: DOI
[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
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