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It is proved that  given $-1/2<s<1/2$, for any $f\in L^2(\mathbb{R})$, there is a unique $u\in\widehat{H}^{|s|}(\mathbb{R})$ such that $$f=\boldsymbol{D}^{-s}u+\boldsymbol{D}^{s*}u\,, $$ where $\boldsymbol{D}^{-s},\boldsymbol{D}^{s*}$ are fractional Riemann-Liouville operators and thefractional derivatives are understood in the weak sense. Furthermore, theregularity of $u$ is discussed, and other versions of the results areestablished. As an interesting consequence, the Fourier transform of elementsof $L^2(\mathbb{R})$ is characterized.

Symmetric Decompositions of $f\in L^2(\mathbb{R})$ Via Fractional Riemann-Liouville Operators