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Continuous and periodic

Let f be a continuous real-valued function on \mathbb{R} satisfying

    \[\left| f(x) \right| \leq \frac{1}{1+x^2} \quad  \forall x\]

Define a function F on \mathbb{R} by

    \[F(x) = \sum_{n=-\infty}^{\infty} f \left ( x + n \right )\]

  1. Prove that F is continuous and periodic with period 1.
  2. Prove that if G is continuous and periodic with period 1 then

        \[\int_{0}^{1} F(x)G(x) \, \mathrm{d}x = \int_{-\infty}^{\infty} f(x) G(x) \, \mathrm{d}x\]

Solution

  1. We note that

        \[g\left ( x + 1 \right ) = \sum_{n=-\infty}^{\infty} f \left ( n +x +1 \right ) = \sum_{n'=-\infty}^{\infty} f \left ( x + n \right ) = g(x)\]

  2. First of all G is bounded on [0, 1] and \sum \limits_{n=-N}^{N} f (x + n ) \rightarrow F(x) uniformly. Hence,

        \begin{align*} \int_{-\infty}^{\infty} f(x) G(x)\, \mathrm{d}x &= \sum_{n=-\infty}^{\infty} \int_{n}^{n+1} f(x) G(x) \, \mathrm{d}x \\ &= \sum_{n=-\infty}^{\infty} \int_{0}^{1} f(x + n) G(x)\, \mathrm{d}x \\ &= \int_{0}^{1} F(x) G(x) \, \mathrm{d}x \end{align*}

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