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- W2023669912 abstract "We define the space of generalized functions $exp _p mathcal{A}'$, $p in N$. If $f in exp _p mathcal{A}'$, then f can be uniquely expanded into a series of the form $( * )qquad f = sum_{n = 0}^infty {a_n psi _n } ,$ where $(psi _n )_{n = 0}^infty $ is an orthonormal base of the corresponding space $L^2 (I)$, I is an interval in R, ${ {b_n } }_{n = 0}^infty $ is a sequence of complex numbers, such that for some $k in N$, $( * * )qquad b_n = Oleft( {left( {underbrace {exp exp cdots exp }_ptilde lambda _n } right)^k } right),quad n = 0,1, cdots ;$${ {lambda _n } }_{n = 0}^infty $ is a sequence of eigenvalues of a corresponding operator $mathcal{R}(mathcal{R}psi _n = lambda _n psi _n )$, and $tilde lambda _n = | {lambda _n } |$, if $lambda _n ne 0$ and $tilde lambda _n = 1$ if $lambda _n = 0$. The series in $( * )$ converges in the sense of weak topology in $exp _p mathcal{A}'$. Conversely, if a sequence ${ {b_n } }_{n = 0}^infty $ satisfies $( * * )$ for some $k in N$, a unique element in $exp _p mathcal{A}'$ is defined by the series in $( * )$. We give representation theorems for elements from $exp _p mathcal{A}'$, $p = 1,2, cdots $, which show that spaces $exp A;exp _2 mathcal{A}', cdots $ are natural generalizations of the space $A'$ from [8]." @default.
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- W2023669912 date "1986-03-01" @default.
- W2023669912 modified "2023-09-25" @default.
- W2023669912 title "Generalization of Zemanian Spaces of Generalized Functions which Have Orthonormal Series Expansions" @default.
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- W2023669912 doi "https://doi.org/10.1137/0517037" @default.
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