Motivated by applications in magnetic resonance relaxometry, we consider the following problem: Given samples of a function $t\mapsto \sum_{k=1}^K A_k\exp(-t\lambda_k)$, where $K\ge 2$ is an integer, $A_k\in\mathbb{R}$, $\lambda_k>0$ for $k=1,\cdots, K$, determine $K$, $A_k$'s and $\lambda_k$'s. Our approach is to transform this function into another function of the same form where $\lambda_k$'s are replaced by $i\lambda_k$. For this purpose, we study the least square approximation using polynomials weighted by the Gaussian weight, and use the fact that Hermite functions are eigenfunctions of the Fourier transform. We provide a detailed analysis of the effect of noise in the data.

翻译：受磁共振弛豫测量应用的启发，我们考虑以下问题：给定函数 $t\mapsto \sum_{k=1}^K A_k\exp(-t\lambda_k)$ 的采样值，其中 $K\ge 2$ 为整数，$A_k\in\mathbb{R}$，$\lambda_k>0$，$k=1,\cdots, K$，确定 $K$、$A_k$ 及 $\lambda_k$。我们的方法是将该函数转换为具有相同形式但 $\lambda_k$ 替换为 $i\lambda_k$ 的另一函数。为此，我们研究了使用高斯权重多项式的最小二乘逼近，并利用埃尔米特函数是傅里叶变换本征函数的性质。我们提供了数据中噪声影响的详细分析。