4.I.8BComplex MethodsPart IB, 2002Let fff be a function such that ∫−∞+∞∣f(x)∣2dx<∞\int_{-\infty}^{+\infty}|f(x)|^{2} d x<\infty∫−∞+∞∣f(x)∣2dx<∞. Prove that∫−∞+∞f(x+k)f(x+l)‾dx=0 for all integers k and l with k≠l,\int_{-\infty}^{+\infty} f(x+k) \overline{f(x+l)} d x=0 \quad \text { for all integers } k \text { and } l \text { with } k \neq l,∫−∞+∞f(x+k)f(x+l)dx=0 for all integers k and l with k=l,if and only if∫−∞+∞∣f^(t)∣2e−imtdt=0 for all integers m≠0\int_{-\infty}^{+\infty}|\widehat{f}(t)|^{2} e^{-i m t} d t=0 \quad \text { for all integers } m \neq 0∫−∞+∞∣f(t)∣2e−imtdt=0 for all integers m=0where f^\widehat{f}f is the Fourier transform of fff.