Cardinal Fourier Transform: A Dimensionally Separated Alternative to the FFT

Standard multidimensional Fast Fourier Transform (FFT) algorithms assume that patterns repeat at boundaries, which can distort momentum measurements in isolated quantum systems. We improve the accuracy of converting spatial data to momentum data by correctly using the sincs functions, while maintaining a format that breaks calculations into manageable parts. We introduce two ideas: one to switch from spatial sincs to momentum sincs, and another operator that multiplies a set of sinc-based functions by the Fourier transform of two sincs. By reworking this change of basis as a series of simple 1D matrix calculations, we keep very high accuracy in multiple dimensions. The operator uses a special momentum formula involving a commutator to account for how two sincs combine. Our new approach shows accurate results for complex rotations, connects systems with or without repeating patterns, speeds up computations in higher dimensions by removing large memory transfers, and greatly reduces total memory requirements.