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 .
 None
Given options, please tell me the categories of all the listed entity words.Output format is "type1: word1; type2: word2". 
Option: location, person, organization 
Text: {0} 
 Entities:  
Answer:
 None