Saturation Theory in Connection with Mellin Transform Methods

Abstract

If X denotes one of the spaces $L^p $, $1 \leqq p < \infty $, or C, where $L^p $ is the set of all f with $[\int _0^\infty | {f(r)} |^p r^{ - 1} dr]^{{1 / p}} < \infty $ and C the set of all f bounded and continuous on $(0,\infty )$ with $\lim _{a \to 1} [\sup _{0 < r < \infty } |f(ar) - f(r)|] = 0$, then $f \in X_{\sigma _1 ,\sigma _2 } $ provided $r^\sigma f(r) \in X$ for every $\sigma \in (\sigma _1 ,\sigma _2 )$. For $f \in X_{\sigma _1 ,\sigma _2 } $ a general approximation process $I_\vartheta (f;r) = \int _0^\infty f({r / \rho })k_\vartheta (\rho )\rho ^{ - 1} d\rho $ of Mellin convolution type is considered, the kernel $\{ k_\vartheta (r)\} $, $\varphi > 0$,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 $f = 0$.