Explicit Integral Transform Proofs of Some Transplantation Theorems for the Hankel Transform
Explicit Integral Transform Proofs of Some Transplantation Theorems for the Hankel Transform
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Hankel变换的一些移植定理的显式积分变换证明
DOI：
10.1137/0504033
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发表时间：
1973
影响因子：
2
通讯作者：
S. Schindler
中科院分区：
文献类型：
--
作者：
S. Schindler
The transplantation theorem for Hankel transforms states that \[ \int_0^\infty {\left| {g_\mu (x)} \right|^p x^{\alpha p} dx \cong } \int_0^\infty {\left| {g_\nu (x)} \right|^p x^{\alpha p} dx} , \] where \[g_\mu (x) = \int ^\infty G(z)(xz)^{{1 / 2}} J_\mu (xz)dz,\quad mu ,\nu \geqq - \tfrac{1}{2},\quad 1 < p < \infty ,\quad {\text{and}}\quad {{ - 1} / p} < \alpha < 1 - {1 / p}.\] It was proved, in a very elegant way, by D. L. Guy in 1960. His proof is, however, an indirect one. It does not completely illuminate (i) the role played by the singular integral transform which arises, (ii) for what values of $\mu $, $\nu $ the singularity disappears, and (iii) when one may expect an $L^1 $ theorem. Our proof explicitly gives the relationship, via an integral transform, between $g_\mu (x)$ and $g_\nu (x)$, and immediately answers these questions. To find the kernel of this integral transform, we first suppose $g_\nu $ is a “good” function. The Weber–Schaf heitlin formula is used and in the course of justifying ...
The transplantation theorem for Hankel transforms states that \[ \int_0^\infty {\left| {g_\mu (x)} \right|^p x^{\alpha p} dx \cong } \int_0^\infty {\left| {g_\nu (x)} \right|^p x^{\alpha p} dx} , \] where \[g_\mu (x) = \int ^\infty G(z)(xz)^{{1 / 2}} J_\mu (xz)dz,\quad mu ,\nu \geqq - \tfrac{1}{2},\quad 1 < p < \infty ,\quad {\text{and}}\quad {{ - 1} / p} < \alpha < 1 - {1 / p}.\] It was proved, in a very elegant way, by D. L. Guy in 1960. His proof is, however, an indirect one. It does not completely illuminate (i) the role played by the singular integral transform which arises, (ii) for what values of $\mu $, $\nu $ the singularity disappears, and (iii) when one may expect an $L^1 $ theorem. Our proof explicitly gives the relationship, via an integral transform, between $g_\mu (x)$ and $g_\nu (x)$, and immediately answers these questions. To find the kernel of this integral transform, we first suppose $g_\nu $ is a “good” function. The Weber–Schaf heitlin formula is used and in the course of justifying ...