Problem 1801 (difficulty: 9/10)

Construct a kernel function \(\displaystyle \varphi:\R\times(0,\infty)\times\R\to(0,\infty)\) with the following property: whenever a function \(\displaystyle h(x,y)\) is harmonic and bounded in the interior of the upper half-plane and it is continuous on the closed half-plane then

\(\displaystyle h(x,y) = \int_{-\infty}^\infty h(t,0) \, \varphi(x,y,t) \dt \)

holds for every \(\displaystyle y>0\).

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