On the Dual Boundary Conditions and the Nonnegative Square Root of the Second Derivative Operator

A dual relationship between the boundary conditions of the second derivative operatorAu=−u″(•) is established. Some dual properties determined by the dual boundary conditions are considered. Furthermore, the singularities of the nonnegative square root of the above second derivative operator are studied. Our results show that, although A1/2u is not u′(•), it is a harmonic conjugate of u′(•) in some sense