A general formulation of shift-invariant "best-basis" expansions is presented. Specifically, we construct an extended library of smooth local trigonometric bases, and introduce a suitable "best-basis" search algorithm. We prove that the resultant decomposition is shift-invariant, orthonormal and characterized by a reduced information cost. The shift-invariance is derived from an adaptive relative shift of expansions in distinct resolution levels. We show that at any resolution level £ it suffices to examine and select one of two relative shift options - a zero shift or a 2-l-1 shift. A variable folding operator, whose polarity is locally adapted to the parity properties of the signal, extra enhances the representation.