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Fundam. Prikl. Mat., 1995, Volume 1, Issue 4, Pages 939–951 (Mi fpm130)  

Convergence exponent of singular integral in generalized Hilbert–Kamke problem

A. Zrein


Abstract: In this article we find exact value of the convergence exponent of singular integral in the problem of simultaneous representation of increasing set of natural numbers $N_1,\ldots,N_r$ by sum of terms $[x^{n_1+\theta}],[x^{n_2+\theta}],\ldots,[x^{n_r+\theta}]$ ($n_1<n_2<\ldots<n_r$ — natural numbers, $0\leq\theta\leq1$). We consider integral:
$$ \theta_0=\int\limits_{\mathbb R^r}|I(\alpha_1,\ldots,\alpha_r)|^k d\alpha_1\ldots d\alpha_r, $$
where $k$ is an unrestricted index and
$$ I(\alpha_1,\ldots,\alpha _r)=\int\limits_{0}^{1}\exp\{2\pi i\sum_{j=1}^{r}\alpha_jx^{n_j+\theta}\} dx. $$
It is proved that $\theta_0$ converges when $k>k_0$ and diverges when $k\leq k_0$ where
$$ k_0=\max \{n_1+\cdots+n_r+r\theta,\frac{r(r+1)}{2}+1\}. $$


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Bibliographic databases:
UDC: 511.336.6
Received: 01.03.1995

Citation: A. Zrein, “Convergence exponent of singular integral in generalized Hilbert–Kamke problem”, Fundam. Prikl. Mat., 1:4 (1995), 939–951

Citation in format AMSBIB
\Bibitem{Zre95}
\by A.~Zrein
\paper Convergence exponent of singular integral in generalized Hilbert--Kamke problem
\jour Fundam. Prikl. Mat.
\yr 1995
\vol 1
\issue 4
\pages 939--951
\mathnet{http://mi.mathnet.ru/fpm130}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1791621}
\zmath{https://zbmath.org/?q=an:0874.11057}


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