The future can , of course , be different . To compute an expansion it would look like 2N terms each with 2N multiplications and additions , hence something like (2N )2 operations of multiplication and addition . But using both (1 ) the addition and subtraction of terms with the same multiplier before doing the multiplications , and (2 ) producing higher frequencies by multiplying lower ones , the Fast Fourier Transform (FFT ) has emerged requiring about N log N operations . This reduction in computing effort has greatly...











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