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- W2069687788 abstract "The Hardy operator $T_a$ on a tree $G$ is defined by [(T_af)(x):=v(x) int^x_a u(t)f(t) dt qquad {for} a, xin G. ] Properties of $T_a$ as a map from $L^p(G)$ into itself are established for $1le p le infty$. The main result is that, with appropriate assumptions on $u$ and $v$, the approximation numbers $a_n(T_a)$ of $T_a$ satisfy [ (*) lim_{nto infty} na_n(T_a) = alpha_pint_{G} |uv|dt ] for a specified constant $alpha_p$ and $1<p<infty$. This extends results of Naimark, Newman and Solomyak for $p=2$. Hitherto, for $pneq 2$, (*) was unknown even when $G$ is an interval. Also, upper and lower estimates for the $l^q$ and weak-$l^q$ norms of ${a_n(T_a)}$ are determined." @default.
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- W2069687788 date "2001-09-01" @default.
- W2069687788 modified "2023-09-25" @default.
- W2069687788 title "The Approximation Numbers of Hardy-Type Operators on Trees" @default.
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- W2069687788 doi "https://doi.org/10.1112/plms/83.2.390" @default.
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