  A Fourier transformation in a fractional dimensional space of order $\la$
($0<\la\leq 1$) is defined to solve the Schr\"odinger equation with Riesz
fractional derivatives of order $\a$. This new method is applied for a particle
in a fractional $\delta$-potential well defined by $V(x) =-
\gamma\delta^{\la}(x)$, where $\gamma>0$ and $\delta^{\la}(x)$ is the
fractional Dirac delta function. A complete solutions for the energy values and
the wave functions are obtained in terms of the Fox H-functions. It is
demonstrated that the eigen solutions are exist if $0< \la<\a$. The results for
$\la= 1$ and $\a=2$ are in exact agreement with those presented in the standard
quantum mechanics.
