# Determining the minimum value

Function $W(t,x)$ is defined as $$W(t,x)=\sum_{i}\alpha_i e^{-\beta_i(t-x)},$$ where $\alpha_i$ is real and $\beta_i$ is real and positive, Then $\Psi$ is defined as $$\Psi =\int_{a}^{b}\Big{(}h(t) -\int_{\lambda_L}^{\lambda_H}W(t,x)\,dx\Big{)}^2\,dt$$ Function $h(t)$ is known. I would like to know (i) is there a set of $\alpha_i-\beta_i$ pairs that produces a minimum for $\Psi$? and (ii) if the minimum exists, is the set of $\alpha_i-\beta_i$ pairs that produce the minimum unique?

You will either want to limit the range of $i$, apply a penalty function for the number of pairs, and/or use some sort of maximum likelihood criterion to decide how many components to decompose into. Without this, your problem is ill-posed (has discontinuous swings of parameters with miniscule changes in your function to be fit).
According to your definitions, you have $$\Psi =\int_{a}^{b}\Big{(}h(t)-\sum_{i=1}^{i=n}\frac{\alpha_i e^{-\beta_i t} \left(e^{\beta_i \lambda_H }-e^{\beta_i \lambda_L }\right)}{\beta_i}\Big)^2$$ and you want to minimize $\Psi$ adjusting the values of parameters $\alpha_i$ and $\beta_i$. This is just an extension of a nonlinear least square fit. Probably the easiest way would be to compute all partial derivatives of $\Psi$ with respect to the $2n$ parameters and force them to be equal to zero (this leads to something very similar to the so called normal equations in the case of linear regression).