The powers $$\delta ^r$$ and $$(\delta ')^r$$ of Dirac-delta function and its derivative for arbitrary real number order are recently redefined in distributional sense by means of fractional derivative, [16, 23]. In this paper, we define the expression $$\delta ^k(f(x))$$ for an infinitely differentiable function f(x) having distinct simple roots and $$k\in {\mathbb N}$$ , and furthermore use the double neutrix limit, due to Fisher, of the regular sequences $$\left[ (\delta _m(f(x))^{-k} \right] _n$$ to interpret the symbol $$\delta ^{-k}(f(x))$$ in which f is an infinitely differentiable function having a simple root.

中文翻译：

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通过 Fisher 方法解释与 Dirac delta 函数相关的一些分布成分

Dirac-delta 函数的幂 $$\delta ^r$$ 和 $$(\delta ')^r$$ 及其对任意实数阶的导数最近通过分数导数在分布意义上重新定义，[16, 23]。在本文中，我们为具有不同简单根和 $$k\in {\mathbb N}$$ 的无限可微函数 f(x) 定义表达式 $$\delta ^k(f(x))$$ ，并且此外，使用正则序列 $$\left[ (\delta _m(f(x))^{-k} \right] _n$$ 的正则序列 $$\left[ (\delta _m(f(x))\right] _n$$ 的双中性极限来解释符号 $$\delta ^ {-k}(f(x))$$ 其中 f 是一个具有简单根的无限可微函数。