Frame decomposition can be delicate. In a separable Hilbert spaces, frames stand for a sequence {x n } ? H such that ?x ? H, A ?x ? 2 ? ? |?x, x n ?| 2 ?B ?x ? 2 , where constants A, B 0 are the frame bounds. The upper bound of the frame is also known as the Bessel condition for {x n }. This frame is tight if A = B and is exact if it ceases to be a frame when any of its elements is removed. There are generally infinitely many duals for a given nonexact frame. Let {x n } be frame for H. Then there is dual {x n } such that for every x in H, x = ? n?x, x n ?x n = ? n ? x, x n ? x n .When H be a separable Hilbert space, we say {x n } and { x n } form a pair of pseudoframes for H if for every x, y in H, ?x, y? = n?x, x n ??x n , y ?. this equation can be particularly convenient for arguments of (weak) convergence that maybe necessary in general cases. Frames are all pseudoframes. However, a pseudoframe pair need not be the usual frames. In this dissertation, a notion of pseudoframes for suaces (PFFS) is deffined and characterized in a separable Hilbert space H. PFFS functions in a manner of a frame for a suace X in H. Yet none of the pair of sequences {x n } and {x n } is necessarily contained in X . This gives rise to attractive properties that center around the flexibility. A necessary and sufficient characterization of PFFSs is provided. Analytical formulae for the constructions of PFFSs in two directions are derived. Examples are considered. Some insight relationships of an PFFS for X and a frame of X are also observed thanks to a private communication with Casazza. So we show that there are duals {x n } to a given (nonexact) frame {x n } that are not usual frames. In particular, these duals are not Bessel sequences. We call them pseudo duals. A characterization and a constructions of pseudo dual are given. In section 4 one example of pseudo dual of a wavelet frame is carefully studied. This was our motivation for the study. We hope it will demonstrate the roles of pseudo duals in theoretical and practical application. To characterize pseudo duals, we introduce a notion of pseudoframes for separable Hilbert spaces in section 5. Basic properties of pseudoframes are discussed. A characterization and a construction of pseudo duals are given. Results are constructive. Example and potential applications of pseudo frames and pseudo duals theory in General Deconvolution Problems and Signal Restorations and Noise Reductions are discussed. We show that if H be a Hilbert space and X be a closed suace of H containing signals of interests, where f is a original signal function and g is a observed signal function with noise n we can restorated signal with flexibility properties of pseudo frames. In fact we hope that f = g and n is removed.