The generalized projection methods in countably normed spaces

Let E be a Banach space with dual space $E^{*}$ E ∗ , and let K be a nonempty, closed, and convex subset of E. We generalize the concept of generalized projection operator “$\Pi _{K}: E \rightarrow K$ Π K : E → K ” from uniformly convex uniformly smooth Banach spaces to uniformly convex uniformly smooth countably normed spaces and study its properties. We show the relation between J-orthogonality and generalized projection operator $\Pi _{K}$ Π K and give examples to clarify this relation. We introduce a comparison between the metric projection operator $P_{K}$ P K and the generalized projection operator $\Pi _{K}$ Π K in uniformly convex uniformly smooth complete countably normed spaces, and we give an example explaining how to evaluate the metric projection $P_{K}$ P K and the generalized projection $\Pi _{K}$ Π K in some cases of countably normed spaces, and this example illustrates that the generalized projection operator $\Pi _{K}$ Π K in general is a set-valued mapping. Also we generalize the generalized projection operator “$\pi _{K}: E^{*} \rightarrow K$ π K : E ∗ → K ” from reflexive Banach spaces to uniformly convex uniformly smooth countably normed spaces and study its properties in these spaces.

New Generalized Projection Speeds Up Audio Declipping

In theory and applications, it is often inevitable to work with projectors onto convex sets, where a linear transform is involved. In this article, a novel projector is presented, which generalizes previous results in that it admits a broader family of linear transforms, but on the other hand it is limited to box-type convex sets in the transformed domain. The new projector has an explicit formula and it can be interpreted within the framework of proximal optimization. The benefit of the new projector is demonstrated on an example from signal processing, where it was possible to speed up the convergence of a signal declipping algorithm by a factor of more than two.

Complete characterisation of attosecond SXR pulses generated by MIR laser sources

Attosecond streaking with broadband SXR continua leads to contributions from multiple overlapping lines in the photoelectron spectrum. The Volkov-transform generalized projection algorithm (VTGPA) is generalised to include all contributing photoelectron bands (multi-line VTGPA) for the reconstruction of ultra-broadband SXR continua. We further investigate the influence of the collection angle of photoelectron detectors on attosecond streaking spectrograms and show full reconstruction for angle-integrated streaking traces. Also, the effects of the photoionization dipole matrix elements on the reconstruction are demonstrated.