{'fraction_non_alphanumeric': 0.013856812933025405, 'fraction_numerical': 0.0, 'mean_word_length': 5.575757575757576, '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}}
A Sinister View of Dilation Equations

We present a technique for studying refinable functions which are compactly supported. Refinable functions satisfy dilation equations and this technique focuses on the implications of the dilation equation at the edges of the support of the refinable function. This method is fruitful, producing new results regarding existence, uniqueness, smoothness and rate of growth of refinable functions.