Saturation Theory in Connection with Mellin Transform Methods
W. Kolbe, R. J. Nessel · SIAM Journal on Mathematical Analysis · 1972
If X denotes one of the spaces $L^p $, $1 \leqq p 0$,satisfying suitable conditions such that it constitutes an approximate identity for $\vartheta \to 0 + $, i.e., \[ \mathop {\lim }\limits_{\vartheta \to 0 + } \left\| {r^\sigma [I_\vartheta (f;r) - f(r)]} \right\|_x = 0\quad {\text{for each }}\sigma \in (\sigma _1 ,\sigma _2 ), f\in X_{\sigma _1,\sigma _2 }.\] The main purpose of this paper is the study of saturation phenomena of the process $I_\vartheta (f;r)$. Using Butzer’s integral transform method, a complete saturation theorem is obtained under suitable conditions upon the Mellin transform of $k_\vartheta (r)$. As a significant application of the general results, the boundary behavior of the solution $u(r,\vartheta )$ of Dirichlet’s problem for the wedge $W = \{ (r,\vartheta )\mid 0 < r < \infty ,0 < \vartheta < \vartheta _0 \} $, $0 < \vartheta _0 < 2\pi $, is considered in detail. This problem is first raised in a strong sense (essentially as an abstract Cauchy problem) which allows a rigorous treatment of its solution via the classical Mellin transform method. In particular, boundary values $f_1 ,f_2 \in X_{ - \sigma _0 ,\sigma _0 } = {\pi / {\vartheta _0 }}$, are attained in the sense that for each $\sigma \in ( - \sigma _0 ,\sigma _0 )$, \[\mathop {\lim }\limits_{\vartheta \to 0 + } \left\| {r^\sigma \left[ {u(r,\vartheta ) - f_1 (r)} \right]} \right\|_X = 0,\quad \mathop {\lim }\limits_{\vartheta \to \vartheta _0 - } \left\| {r^\sigma \left[ {u(r,\vartheta ) - f_2 (r)} \right]} \right\|_X = 0.\] For, for example, symmetric boundary values $f_1 = f_2 = f$, the general saturation theorem then gives that the above quantities cannot tend to zero too rapidly; thus $\| {r^\sigma [u(r,\vartheta ) - f(r)]} \|_X = o(\vartheta )$, $\vartheta \to 0 + $, implies $f = 0$.