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Function $W(t,x)$ is defined as \begin{equation} W(t,x)=\sum_{i}\alpha_i e^{-\beta_i(t-x)}, \end{equation} where $\alpha_i$ is real and $\beta_i$ is real and positive, Then $\Psi$ is defined as \begin{equation} \Psi =\int_{a}^{b}\Big{(}h(t) -\int_{\lambda_L}^{\lambda_H}W(t,x)\,dx\Big{)}^2\,dt \end{equation} 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?

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I would start with On Least Squares Exponential Sum Approximation With Positive Coefficients. Then go on to On Approximation of Functions By Exponential Sums and its follow-on paper Approximation by exponential sums revisited.

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).

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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).

But, since the model is highly nonlinear with respect to its parameters, the difficulty will come from the imperious need of "reasonable" guesses for all parameters.

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