Spectral representation of the weighted Laplace transform


We find the spectral representation of the selfadjoint operators T Tf(λ)≔∫0∞K(λt)f(t)dt,in L2(]0,∞[), where 0≤K∈Lloc1(0,∞). More precisely (see Theorem 4.1) for these operators which include the Laplace transform as a special case, the spectrum of T is a compact interval [−κ,κ], and we find explicitly a unitary operator U:L2(]0,∞[)→L2(R) and a continuous real function α on R such that UTU−1 is the operator of multiplication by α.

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Applied Mathematics Letters