We consider the $\alpha$-sine transform of the form $T_\alpha f(y)=\int_0^\infty\vert\sin(xy)\vert^\alpha f(x)dx$ for $\alpha>-1$, where $f$ is an integrable function on $\mathbb{R}_+$. First, the inversion of this transform for $\alpha>1$ is discussed in the context of a more general family of integral transforms on the space of weighted, square-integrable functions on the positive real line. In an alternative approach, we show that the $\alpha$-sine transform of a function $f$ admits a series representation for all $\alpha>-1$, which involves the Fourier transform of $f$ and coefficients which can all be explicitly computed with the Gauss hypergeometric theorem. Based on this series representation we construct a system of linear equations whose solution is an approximation of the Fourier transform of $f$ at equidistant points. Sampling theory and Fourier inversion allow us to compute an estimate of $f$ from its $\alpha$-sine transform. The same approach can be extended to a similar $\alpha$-cosine transform on $\mathbb{R}_+$ for $\alpha>-1$, and the two-dimensional spherical $\alpha$-sine and cosine transforms for $\alpha>-1$, $\alpha\neq 0,2,4,\dots$. In an extensive numerical analysis, we consider a number of examples, and compare the inversion results of both methods presented.
翻译:我们考虑的是 $\ alpha f(y)\\ int_ 0\ infty\ vert\ sin(xy)\ vert\ alpha f(x) dx$ 美元, $\ ALpha>-1 美元, 美元是无法解释的函数 $\ phab{R\ 美元。 首先, 将这种变换转换换为$\ alpha> 1 美元, 是在一个比较普遍的组合背景下讨论的, 在正数正数线上的加权和可折数函数空间进行整体变换换。 在另一种办法中, 我们显示的是 $\ alpha $ 美元 的变换, 美元 美元 美元 的变换, 相同 的 理论 和 四变 法 法 的 方法, 都可以 以 $ 美元 的变法 。