Optimal noise suppression: A geometric nature of pseudoframes for subspaces

Pseudoframes for subspaces (PFFS) is a notion of frame-like expansions for a subspace $\chi$ in a separable Hilbert space [11]. The spanning nature of the sequences $\{x_n\}$ and $\{x^*_n\}$ in a PFFS (relative to the subspace $\chi$) is generally very different from that of a frame. Incidentally, a PFFS constitutes generally a nonorthogonal projections onto $\chi$. The directions of the projection determine the geometric meanings and its applications of a PFFS. PFFS also provides a means for the construction of nonorthogonal projections that arises in various linear reconstruction problems. This article is aimed at elaborations on such geometrical properties, demonstration of natural needs of nonorthogonal projections in applications and how PFFS can be applied, particularly for optimal noise suppressions. In this specific application, we show that PFFS is not only natural and sufficient but also necessary for generating an optimal solution among the class of all linear and series-based methods.

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