{'fraction_non_alphanumeric': 0.01762114537444934, 'fraction_numerical': 0.0, 'mean_word_length': 5.2, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Mathematical background for generalized, partial, and incomplete discrete Fourier transforms

We develop the theory of generalized discrete Fourier transforms (GDFTs) from the point of view of the Chinese Remainder Theorem (CRT). We give a new definition of GDFT, and apply it to the construction of multidimensional convolution algorithms which require significantly fewer multiplications and data transfer operations than the usual methods. We also show how CRT isomorphisms which are similar to GDFTs can be used in convolution algorithms. These CRT mappings increase the number of values for the length N of a convolution in a ring R which can be done with FFT-like algorithms.