Zeta regularization and product of dirac delta distribution

[zetafunction]

using the convolution theorem with power functions $$x^{m}$$ we may define via the convolution theorem the product of 2 dirac delta distribution

then main idea is to consider the convolution integral $$\int_{R}dt(x-t)^{m}t^{n}$$

and then apply the Fourier transform with respect to variable 'x' (here , and n are positive integers)

of course, this integral will be DIVERGENT for any value of 'x' due to the expressions

$$\int_{R}dt t^{m}$$ (integrals over the Real line)

however by making an extension of the Zeta regularization algorithm for divergent series , we can make sense of integrals of the form

$$\int_{R}dt(x-t)^{m}t^{n}$$

here.. http://vixra.org/abs/1005.0071

we present some examples of product of distributions involving the dirac delta function, Heaviside Step function and finite part (in Cauchy's sense) , we discuss some of the applications of the Zeta regularization algorithm for integrals and how in the limit of finite N (upper limit) we get the expected result of calculus $$(m+1)\int_{0}^{N}x^{m}dx=N^{m+1}$$

 

[Valerie Park]

/ (m+1) in this way we can make sense of the product of the dirac delta distribution and the power function x^{m} , as a result of the convolution theorem the result is that the product of the Dirac Delta Function and the power function x^{m} is equivalent to the finite part of the integral \int_{R}dt(x-t)^{m}t^{n} as obtained via Zeta regularization algorithm