On a class of singular integral equations with translations (Q908469)

Singular integral equations with constant coefficients and with two translations are discussed. These equations are reduced to two boundary value problems (BVP) for analytic functions \(\Phi^+(z)=\lambda (T_{\alpha}\Phi^+)(z)+1/(a+b)(S^+f)(z),\quad y\geq 0;\quad \Phi^- (z)=\mu (T_{\beta}\Phi^-)(z)+1/(a-b)(S^-f)(z),\quad y\leq 0\) with translation operators \((T_{\gamma}g)(z)=g(z+\gamma).\) In the case \(| \lambda | <1\) the unique solution of the first problem is given in series form (analogously for the second one). As for the cases \(| \lambda | =1\) or \(| \lambda | >1\) the method of the BVP's solution is more complicated and uses some properties of entire functions.