We study the conditional integral transform, the conditional convolution product and the first variation of functionals on function space. For our research, we modify the class $\Cal S_{\alpha}$ of functionals introduced in [7]. We then give the existences of the conditional integral transform, the conditional convolution product and the first variation for functionals in $\Cal S_{\alpha}$. Finally, we give various relationships and formulas among conditional integral transforms, conditional convolution products and first variations of functionals in $\Cal S_{\alpha}$.