: 3. If f in L (-∞,∞), show that lim ∫f(x)cos(nx)dx = 0.
Clearly, f(x)cos(nx) is in L1. Let In = ∫f(x)cos(nx)dx
Let y = x - π/n, then
In = ∫f(y+π/n)cos(π+ny)dy = ∫-f(y+π/n)cos(ny)dy.
2|In| = |In + In| = |∫(f(x)-f(x+π/n))cos(nx)dx|
≦ ∫|f(x)-f(x+π/n)| = ∥f(。) - f(。 + π/n)∥ tends to zero
as n tends to infinty since f is in L1.
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