On progressive functions

Progressive functions at time t involve only the progressive functions at time before t and some nice compactly supported function at time t. We give sufficient conditions and explicit formulas to construct progressive functions with exponential decay and characterize the conditions on which the positive integer translates of a progressive function are orthonormal or a Riesz sequence. We provide explicit ways for construction of orthonormal progressive functions and for construction of the biorthogonal functions of nonorthogonal progressive functions. Such progressive functions can be used to construct wavelets with arbitrary smoothness on the half line if they are generated by a smooth refinable compactly supported function.