Let $$\hat{f}(\lambda):= \int_0^{+\infty}K(x,\lambda)\ f(x)\ dx, \text{where } K(x,\lambda)=\sqrt{\frac{2}{\pi}}\frac{\lambda \cos(\lambda x)+h\sin(\lambda x)}{\lambda^2+h^2}$$ be the (Fourier) integral transform of function $f: (0,+\infty)\to\mathbf{R}$, where $h>0$ is a parameter ( a positive constant), what should be the inverse integral transform of $\hat{f}(\lambda)$? Is the inverse (denoted by $ \big(\hat{f}\big)^{\check{~~}} $ be $$f(x)=\big(\hat{f}(\lambda)\big)^{\check{~~}}(x):=\int_0^{+\infty}K(x,\lambda)\ \hat{f}(\lambda)\ d\lambda\ ?$$ This question is based on a problem (the problem number is 64) on Page 60 of Book of B.M.Budak, A. A. Samarskii and A. N. Tikhonov, titled as A Collection of Problems in Mathematical Physics (New York, Dover, 1964). This kind integral transform is useful in solving the heat equation on half line with Robin boundary values. I have no way of figuring out its inverse, so I propose it here, so as someone can help me.
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