Remarks on Convolutions and Fractional Derivative of Distributions

  •  Chenkuan Li    
  •  Kyle Clarkson    


This paper begins to present relations among the convolutional definitions given by Fisher and Li, and further shows that the following fractional Taylor's expansion holds based on convolution \[ \frac{d^\lambda}{d x^\lambda}  \theta (x) \phi(x) = \sum_{k = 0}^{\infty} \frac{\phi^{( k)}(0)\, x_+^{k - \lambda }}{\Gamma(k - \lambda  + 1)} \quad \mbox{if} \quad \lambda \geq 0, \] with demonstration of several examples.  As an application, we solve the Poisson's integral equation below  \[ \int_0^{\pi/2} f(x \cos \omega)\sin^{2 \lambda + 1} \omega d \omega = \theta(x) g(x) \] by fractional derivative of distributions and the Taylor's expansion obtained.

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