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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 \([0,2)\).

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
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