Дается описание безусловных базисов вида $\{d_\alpha(i\lambda_nt):\lambda_n\in\Lambda\}$ в
пространстве $L_2(-a, a)$ с мерой $|x|^\gamma dx$, $\gamma=2\alpha+1$. Здесь ядро Данкла $d_\alpha(ixt)$ определяется с помощью формулы:
$$
d_\alpha(z)=2^\alpha\Gamma(\alpha+1)z^{-\alpha}(J_\alpha(z)+iJ_{\alpha+1}(z)), \qquad \alpha>-1,
$$
где $J_\alpha$ - функция Бесселя первого рода.
The non-dissipative operator of integration is studied in the weight space. Its similarity to the operator of integration in the space without weight is proved. The functional model for this operator is obtained.
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