| dc.contributor.author | Mytskan, Mykhaylo | |
| dc.date.accessioned | 2021-11-30T21:59:54Z | |
| dc.date.available | 2021-11-30T21:59:54Z | |
| dc.date.issued | 2021 | |
| dc.identifier.citation | Zvozdetskyi T. I., Mytskan M. M. On the equivalence of some convolutional equalities in spaces of sequences, Bukovinian Math. Journal. 9, 1 (2021), 180 188. | uk_UA |
| dc.identifier.uri | https://archer.chnu.edu.ua/xmlui/handle/123456789/2912 | |
| dc.description.abstract | The problem of the equivalence of two systems with $n$ convolutional equalities arose in
investigation of the conditions of similarity in spaces of sequences of operators which
are left inverse to the $n$-th degree of the generalized integration operator. In this paper we solve this problem. Note that we first prove the equivalence of two corresponding systems with $n$ equalities in the spaces of analytic functions, and then, using this statement, the main result of paper is received.
Let $X$ be a vector space of sequences of complex numbers with K$\ddot{\rm o}$the normal topology from a wide class of spaces,
${\mathcal I}_{\alpha}$ be a generalized integration operator in $X$, $\ast$ be a nontrivial convolution for ${\mathcal I}_{\alpha}$ in $X$,
and $(P_q)_{q=0}^{n-1}$ be a system of natural projectors with $\displaystyle x = \sum\limits_{q=0}^{n-1} P_q x$ for all $x\in X$.
We established that a set $(a^{(j)})_{j=0}^{n-1}$ with
$$
\max\limits_{0\le j \le n-1}\left\{\mathop{\overline{\lim}}\limits_{m\to\infty} \sqrt[m]{\left|\frac{a_{m}^{(j)}}{\alpha_m}\right|}\right\}<\infty
$$
and a set $(b^{(j)})_{j=0}^{n-1}$ of elements of the space $X$ satisfy the system of equalities
$$
b^{(j)}=a^{(j)}+\sum\limits_{k=0}^{n-1}({\mathcal I}_{\alpha}^{n-k-1} a^{(k)}) \ast {(P_{k}b^{(j)})}, \quad j = 0, 1, ... \, , \, n-1,
$$
if and only if they satisfy the system of equalities
$$
b^{(j)}=a^{(j)}+\sum\limits_{k=0}^{n-1}({\mathcal I}_{\alpha}^{n-k-1} b^{(k)}) \ast {(P_{k}a^{(j)})}, \quad j = 0, 1, ... \, , \, n-1.
$$
Note that the condition on the elements $(a^{(j)})_{j=0}^{n-1}$ of the space $X$ allows us to reduce the solution of this problem to the solution of an analogous problem in a space of functions analytic in a disc. | uk_UA |
| dc.description.sponsorship | Математичного аналізу | uk_UA |
| dc.language.iso | other | uk_UA |
| dc.subject | згортка | uk_UA |
| dc.subject | простір послідовностей | uk_UA |
| dc.subject | простір аналітичних функцій | uk_UA |
| dc.subject | оператор узагальненого інтегрування | uk_UA |
| dc.subject | convolution | uk_UA |
| dc.subject | space of sequences | uk_UA |
| dc.subject | space of analytic functions | uk_UA |
| dc.subject | generalized integration operator | uk_UA |
| dc.title | Про рівносильність деяких згорткових співвідношень у просторах послідовностей | uk_UA |
| dc.title.alternative | On the equivalence of some convolutional equalities in spaces of sequences | uk_UA |
| dc.type | Article | uk_UA |
