We propose a numerical method to spline-interpolate discrete signals and then apply the integral transforms to the corresponding analytical spline functions. This represents a robust and computationally efficient technique for estimating the Laplace transform for noisy data. We revisited a Meijer-G symbolic approach to compute the Laplace transform and alternative approaches to extend canonical observed time-series. A discrete quantization scheme provides the foundation for rapid and reliable estimation of the inverse Laplace transform. We derive theoretic estimates for the inverse Laplace transform of analytic functions and demonstrate empirical results validating the algorithmic performance using observed and simulated data. We also introduce a generalization of the Laplace transform in higher dimensional space-time. We tested the discrete LT algorithm on data sampled from analytic functions with known exact Laplace transforms. The validation of the discrete ILT involves using complex functions with known analytic ILTs.

翻译：我们提出了一种数值方法，对离散信号进行样条插值，并在相应的解析样条函数上应用积分变换。这是一种稳健且计算高效的技术，用于估计含噪数据的拉普拉斯变换。我们重新探讨了Meijer-G符号化方法来计算拉普拉斯变换，并提出了扩展典型观测时间序列的替代方案。一种离散量化方案为快速可靠地估计逆拉普拉斯变换提供了基础。我们推导了解析函数的逆拉普拉斯变换的理论估计，并利用观测数据和模拟数据展示了验证算法性能的实证结果。我们还引入了高维时空中的拉普拉斯变换的推广形式。我们在从具有已知精确拉普拉斯变换的解析函数采样的数据上测试了离散LT算法。离散ILT的验证涉及使用具有已知解析ILT的复函数。