Let $C[0,t]$ denote the function space of all real-valued continuous paths on $[0,t]$. Define $X_n:C[0,t]{ ightarrow}mathbb{R}^{n+1}$ by $Xn(x)=(x(t_0),x(t_1),{cdots},x(t_n))$, where $0=t_0$ < $t_1$ < ${cdots}$ < $t_n$ < $t$ is a partition of $[0,t]$. In the present paper, using a simple formula for the conditional expectation given the conditioning function $X_n$, we evaluate the $L_p(1{leq}p{leq}{infty})$-analytic conditional Fourier-Feynman transform and the conditional convolution product of the cylinder functions which have the form $$f((v_1,x),{cdots},(v_r,x));for;x{in}C[0,t]$$, where {$v_1,{cdots},v_r$} is an orthonormal subset of $L_2[0,t]$ and $f{in}L_p(mathbb{R}^r)$. We then investigate several relationships between the conditional Fourier-Feynman transform and the conditional convolution product of the cylinder functions.