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 derivativ e operator Au = -u "(.) is established. Some dual properties determined by the dual boundary condit ions are considered. Furthermore, the singularities of the nonnegative squa re root of the above second derivative operator are studied. Our results sh ow that, although A(1/2)u is not u'(.), it is a harmonic conjugate of u'(.) in some sense. (C) 2000 Academic Press.