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psie

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- Homework Statement
- Prove directly that if $$K_n(s)=\begin{cases}n &\text{if }|s|<1/(2n)\\ 0 &\text{if }|s|>1/(2n),\end{cases}$$ and ##f## is a continuous function at the origin, then $$\lim_{n\to\infty}\int_\mathbb{R}K_n(s)f(s)ds=f(0).$$

- Relevant Equations
- Positive summability kernels, see e.g. Wikipedia.

This is an exercise from Fourier Analysis and its Applications by Vretblad.

I know the integral over ##\mathbb R## reduces to $$\int_{-1/(2n)}^{1/(2n)} nf(s)ds.$$ But I don't know where to go from here. There is a theorem in the book which states that this limit exists and equals ##f(0)##, but I'm apparently not supposed to use the theorem. Appreciate any help.

I know the integral over ##\mathbb R## reduces to $$\int_{-1/(2n)}^{1/(2n)} nf(s)ds.$$ But I don't know where to go from here. There is a theorem in the book which states that this limit exists and equals ##f(0)##, but I'm apparently not supposed to use the theorem. Appreciate any help.