Pseudoframes and non-orthogonal projections onto subspaces (Q2839794)

The paper considers a recently introduced class of objects, namely pseudoframes for subspaces in Hilbert spaces (PFSF), derived from the well known class of frames for Hilbert spaces, a basic tool in wavelet theory. The frames in a Hilbert space are based on the existence of a dual object, the (direct) frame and the (dual) frame of the first. In this approach, a signal (subject to some constraints) can be represented by two alternate equivalent forms, each of them being a bilinear form in the elements of the two frames. The coefficients of one representation come from the scalar product between the signal and the elements of the element of one frame whereas the sum is carried on the elements of the other frame. Also, the sum of the squared coefficients is bounded from below and from above, by the square of the signal norm weighted by some positive constants, the so-called frame bounds. Here, in the PFSF approach, one retains one half of the essential properties of the usual frame approach: the representation of the signal makes use of one representation and the two bound inequalities are reduced to a single one. This is one where the squared sum is bounded from above. This constitutes a Bessel-line inequality. With these new specific properties, the authors define some properties which are the corresponding ones to the properties and concepts encountered in the usual frame theory, such as: dual pseudo frame, pseudo frame operator, etc. The authors underline the new facts that are present in the PFSF approach, specifically, the approximation of the signal and the non-orthogonal projection on the null space on some direction. The specific results obtained by the authors are given in Theorems 5 to 9. Although of small extent, the paper is rich in content. It is interesting to see in the following if this new approach will lead to some new point of view.