Transmutation operators and Paley-Wiener theorem associated with a differential-difference operator on the real line (Q2721637)

On the real line, the authors consider the difference-differential operator NEWLINE\[NEWLINE\Lambda f=df/dx+2^{-1}(f(x)-f(-x))A'(x)/A(x)NEWLINE\]NEWLINE (here \(A(x)=| x| ^{2\alpha +1}B(x),\;\alpha >-1/2\), and \(B\) is an even and strictly positive \(C^{\infty}\)-function on \textbf{R}). A pair of integral operators intertwining \(\Lambda\) and \(d/dx\) is constructed explicitely. With the help of this pair, a Plancherel-type theorem related to \(\Lambda\) is proved.