379123
Proof : Substituting ( 2 ) into the right-hand-side of ( 3 ) , we get This is exactly equal to v_j , because \ sum _ ^ \ alpha ^ = 0 when j' \ neq j ( by ( 1 ) with k = j' - j ) , and \ sum _ ^ \ alpha ^ = n when j' = j. ∎ Matrix formulation Since the discrete Fourier transform is a linear operator , it can be described by matrix multiplication .
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Please list all entity words in the text that fit the category. Output format is "word1, word2". 
Option: location, person, organization 
Text: {0} 
Answer:
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