Abs[x]

Piecewise[{{-x,x<0},{x,x>=0}}]

f[r_]:= 1/16 ((-3+6r) Abs[1 - 2 r] + (3 - 2 r) Abs[3 - 2 r]
- 3 (1 + 2 r) Abs[1 + 2 r] + (3 + 2 r) Abs[3 + 2 r])

Assuming[r > 1/2, Simplify[f[r]]]
Assuming[-1/2 < r < 1/2, Simplify[f[r]]]
Assuming[r < -1/2, Simplify[f[r]]]
Assuming[-2 < r < 2, Simplify[f[r]]]
Assuming[-1/2 < r < 3/2, Simplify[f[r]]]
Assuming[0 < r < 2, Simplify[f[r]]]

Out[2]=

Out[3]=

Out[4]=

Out[5]=1/16 ((-3 + 6 r) Abs[1 - 2 r] + (3 - 2 r) Abs[3 - 2 r]
 - 3 (1 + 2 r) Abs[1 + 2 r] + (3 + 2 r) Abs[3 + 2 r])

 Out[6]=

Out[7]=1/16 (6 - 8 r^2 + (-3 + 6 r) Abs[1 - 2 r] + (3 - 2 r) Abs[3 -2 r])

g[n_Integer] := Assuming[-n <= r <= n,
                Integrate[1/(2 [Pi]) (Sin[k/2]/(k/2))^n
                (Cos[k r] - I Sin[k r]),
                {k, -[Infinity], [Infinity]}]]