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I am trying to understand FFT algorithm explained here

FFT(x) {   n=length(x);   
if (n==1) return x;   
m = n/2;   
X = (x_{2j})_{j=0}^{m-1};   
Y = (x_{2j+1})_{j=0}^{m-1};  
X = FFT(X);   
Y = FFT(Y);  
U = (X_{k mod m})_{k=0}^{n-1};   
V = (g^{-k}Y_{k mod m})_{k=0}^{n-1};   
return U+V; 
} 

The author says that the above comes from Danielson-Lanczos Lemma.

I am unable to understand what is the meaning of the lines:

X = (x_{2j})_{j=0}^{m-1};

Y = (x_{2j+1})_{j=0}^{m-1};

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The meaning of the first line is "Let $X$ be an array of length $m$ (with indices running from $0$ to $m-1$) such that $X_j = x_{2j}$ for $j \in \{0,\ldots,m-1\}$." The second line has a similar meaning, with $Y_j = x_{2j+1}$ instead.

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  • $\begingroup$ Thanks. Which langauge is this BTW? $\endgroup$ – gpuguy Apr 19 '13 at 5:16
  • $\begingroup$ It's pseudocode, so no particular language. $\endgroup$ – Yuval Filmus Apr 19 '13 at 14:36
  • $\begingroup$ @gpuguy: perhaps you meant "which language expresses subscript with _ and superscript with ^ and grouping with { }? The answer is TeX/LaTeX, and various informal kinds of ascii math. Put your statements inside $ signs on Stackexchange and it will magically turn into well formatted math! For example $X = (x_{2j})_{j=0}^{m-1}$ yields $X = (x_{2j})_{j=0}^{m-1}$. $\endgroup$ – Wandering Logic Apr 20 '13 at 13:00

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