Let's apply two integral transforms to a function $f(t)$.
$$ F'(v) = \int\limits_{t_1}^{t_2} K_1(t,v) f(t) \mathrm{d}t $$
$$ F(u) = \int\limits_{v_1}^{v_2} K_2(v,u) F'(v) \mathrm{d}v $$
Can I construct a kernel $K(t,u)$ that would replace the two? Like this:
$$ F(u) = \int\limits_{t_1}^{t_2} K(t,u) f(t) \mathrm{d}t $$
I attempted to simply insert it and it seems that it's nearly possible:
$$ F(u) = \int\limits_{v_1}^{v_2} K_2(v,u) \int\limits_{t_1}^{t_2} K_1(t,v) f(t) \mathrm{d}t \, \mathrm{d}v $$
$$ F(u) = \int\limits_{v_1}^{v_2} \int\limits_{t_1}^{t_2} K_1(t,v) K_2(v,u) f(t)\mathrm{d}t \, \mathrm{d}v $$
So I am tempted to say that
$$ K(t,u) = \int\limits_{v_1}^{v_2} K_1(t,v) K_2(v,u) \mathrm{d}v $$
But is that legitimate? Can I change the order of integration that freely here? Usually we have to reconsider the limits of integrations when swapping the order...
I am asking this because I am interested in kernels as of continuous analog of matrices and this "kernel product" would seem to be the analog of matrix multiplication.