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Fourier transform of a triangular function

For

θ_{n} (x) = (1 - ∣ x ∣/ n)_{+}

,

θ_{n} (ξ) = \frac{2 ( 1 - c o s ( n ξ ))}{n ξ ^{2}} = n (\frac{s i n ( n ξ /2 )}{n ξ /2})^{2},

with value

n

at zero.

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Solution