13 Illustration of Formulas Derived From Mellin & Fourier Convolutions

Navigation Links: – First Page – First Child Page

This page is currently under construction, so it might not make total sense at this point in time.

Note that the Fourier series representation of $U(x)$, $U'(x)$, and $U”(x)$ converge to $\theta(x-1)$, $\delta(x-1)$, and $\delta'(x-1)$ in the right-half plane and the Mellin transform $\mathcal{M}_x[\delta(x-1)](s)=1$. I’m currently investigating convergence of the Mellin transforms of the Fourier series representations of $U(x)$, $U'(x)$, and $U”(x)$ and convergence of formulas derived from the Mellin convolutions defined in (1) to (5) below where the Fourier series representation of $U'(x)$ is substituted for $\delta(x-1)$ in the convolutions. Note that convolutions $*_{\mathcal{M}_1}$ and$*_{\mathcal{M}_3}$ are commutative whereas $*_{\mathcal{M}_2}$ is not.

(1) $\quad g(y)=\delta(x-1)\,*_\mathcal{M_1}\,g(x)=\int_0^\infty x^{-1}\,\delta(x-1)\,g\left(\frac{y}{x}\right)\,dx$

(2) $\quad g(y)=g(x)\,*_\mathcal{M_1}\,\delta(x-1)=\int_0^\infty x^{-1}\,g(x)\,\delta\left(\left(\frac{y}{x}\right)-1\right)\,\,dx\,,\quad y>0$

(3) $\quad g(y)=\delta(x-1)\,*_\mathcal{M_2}\,g(x)=\int_0^\infty\delta(x-1)\,g(y\,x)\,\,dx$

(4) $\quad g(y)=\delta(x-1)\,*_\mathcal{M_3}\,g(x)=\int_0^\infty\delta(x-1)\,g(y+1-x)\,dx$

(5) $\quad g(y)=g(x)\,*_\mathcal{M_3}\,\delta(x-1)=\int_0^\infty g(x)\,\delta(y-x)\,\,dx\,,\quad y>0$

I’ve noticed convolutions (1) to (5) above can be normalized as follows which sometimes leads to additional results or the only results when it’s impossible to successfully evaluate the integrals associated with formulas (1) to (5) above. I’ve also noticed $\Re(m)\ge 0$ seems to be useful as well as $\Re(m)<0$.

(6) $\quad g(y)=\delta(x-1)\,*_{\mathcal{M}_{1_n}}\,g(x)=\int_0^\infty x^m\,\delta(x-1)\,g\left(\frac{y}{x}\right)\,dx$

(7) $\quad g(y)=g(x)\,*_\mathcal{M_{1_n}}\,\delta(x-1)=\int_0^\infty x^m\,g(x)\,\delta\left(\left(\frac{y}{x}\right)-1\right)\,dx\,,\quad y>0$

(8) $\quad g(y)=\delta(x-1)\,*_{\mathcal{M}_{2_n}}\,g(x)=\int_0^\infty x^m\,\delta(x-1)\,g(y\,x)\,dx$

(9) $\quad g(y)=\delta(x-1)\,*_\mathcal{M_{3_n}}\,g(x)=\int_0^\infty x^m\,\delta(x-1)\,g(y+1-x)\,dx$

(10) $\quad g(y)=g(x)\,*_\mathcal{M_{3_n}}\,\delta(x-1)=y^{-m}\int_0^\infty x^m\,g(x)\,\delta(y-x)\,\,dx\,,\quad y>0$

I’ve also noticed it’s possible to take derivatives of some functions using Fourier series representations of the derivatives of $\delta(x-1)$ via relationships such as the following.

(11) $\quad g^{(n)}(x)=(-y)^{-n}\left(\delta^{(n)}(x-1)\,*_{\mathcal{M}_2}\,g(x)\right)=(-y)^{-n}\int_0^\infty\delta^{(n)}(x-1)\,g(y\,x)\,dx$

An initial formula derived via one of the convolutions described above can sometimes be integrated or differentiated to derive additional formulas for related functions. When the initial formula converges for $\Re[y]>0$, Laplace and Mellin normal and inverse transforms can also sometimes be used to derive additional formulas for related functions. Formulas for $e^x$ can also sometimes be used to derive formulas for related functions which are defined in terms of integrals containing $e^x$ terms.

I’m currently investigating convergence of a number of new formulas derived from the convolutions described above for a variety of functions such as $x$, $\frac{1}{x+a}$, $log[x]$, $e^x$, $\sin[x]$, $\cos[x]$, $Ei[x]$, $E_n[x]$, $Gamma[a,\,x]$, and various $Bessel$ functions. I plan to illustrate several of these formulas on children pages below when I find time, but right now my priority is derivation of new formulas from the convolutions defined above and investigation of the convergence of these formulas.