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    $\lim_{n\to\infty}\int_{\RR}f(x)g(x+n)\,dx=0$ for $f,g\in L^2(\RR)$

    P-K6CAP

    Area
    Real Analysis
    Topics
    Convergence of Integrals, L², Small Tails
    Seen at
    UGA
    Years
    2016
    Status
    draft

    Let \(f, g \in L^2(\RR)\). Show that \[ \lim _{n \to \infty} \int _{\RR} f(x) g(x+n) \,dx = 0 \]

    • Cauchy Schwarz: \(\norm{fg}_1 \leq \norm{f}_1 \norm{g}_1\).
    • Small tails in \(L^p\).
    Solution

    Let \[ I_n:=\int_{\mathbb R} f(x)g(x+n)\,dx. \] Fix \(\varepsilon>0\). Choose \(h\in L^2(\mathbb R)\) with compact support such that \[ \|g-h\|_2<\frac{\varepsilon}{2(1+\|f\|_2)}. \] Then, by Cauchy–Schwarz and translation invariance, \[ \left|\int f(x)(g-h)(x+n)\,dx\right| \le \|f\|_2\|g-h\|_2<\frac\varepsilon2. \]

    Choose \(R>0\) so that \(h=0\) a.e. outside \([-R,R]\). Then \[ \int f(x)h(x+n)\,dx =\int_{-n-R}^{-n+R}f(x)h(x+n)\,dx, \] so \[ \left|\int f(x)h(x+n)\,dx\right| \le \|h\|_2 \left(\int_{-n-R}^{-n+R}|f(x)|^2\,dx\right)^{1/2}. \] Since \(f\in L^2(\mathbb R)\), its \(L^2\) tails tend to zero, and therefore the last expression tends to \(0\) as \(n\to\infty\). For all sufficiently large \(n\) it is below \(\varepsilon/2\). Hence \(|I_n|<\varepsilon\) eventually, proving \[ \boxed{\lim_{n\to\infty}\int_{\mathbb R}f(x)g(x+n)\,dx=0.} \]

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