Integral transforms of Fourier cosine and sine generalized convolution type (Q925470)

Summary: Integral transforms of the form \[ \begin{multlined} f(x)\mapsto g(x)=(1-d^2/dx^2)\cdot\\ \cdot\Biggl\{\int_0^\infty k_1(y)[f(|x+y-1|)+f(|x-y+1|)-f(x+y+1)-f(|x-y-1|)]\,dy+\\ \int_0^\infty k_2(y)[f(x+y)+f(|x-y|)]\,dy \Biggr\} \end{multlined} \] from \(L_p(\mathbb R_+)\) to \(L_q(\mathbb R_+)\), \((1\leq p\leq 2,\;p^{-1}+q^{-1}=1)\) are studied. Watson's and Plancherel's theorems are obtained.