Algebra of one-dimensional generalized functions

An associative algebra $\mathscr{A}$, equipped with involution and differentiation, is constructed for generalized functions of one variable that at one fixed point can have singularities like the delta function and its derivatives and also finite discontinuities for the function and all its derivatives. The elements of $\mathscr{A}$ together with the differentiation operator form the algebra of local observables for a quantum theory with indefinite metric and state vectors that are also generalized functions. By going over to a smaller space, one can obtain quantum models with positive metric and with strongly singular concentrated potentials.