(FUNDAMENTAL AND APPLIED MATHEMATICS)

1995, VOLUME 1, NUMBER 4, PAGES 939-951

## Convergence exponent of singular integral in generalized Hilbert--Kamke problem

A.Zrein

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 N1,..., Nr by sum of terms $[x^{n_1+\theta}], [x^{n_2+\theta}],\ldots, [x^{n_r+\theta}]$ (n1 < n2 < ... < nr - natural numbers, $0 \leq \theta \leq 1$).

We consider integral:

$\theta _0 = \int_{\mathbb{R}^r} | I (\alpha _1,\ldots, \alpha _r)|^k d\alpha _1\ldots d\alpha _r,$
where k is an unrestricted index and
$I (\alpha _1,\ldots, \alpha _r) = \int_0^1 \exp \{ 2\pi i \sum_{j=1}^r \alpha_j x^{n_j+\theta} \} dx.$
It is proved that $\theta _0$ converges when k>k0 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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