The discrete Cesàro operator C is investigated in the class of smooth sequence spaces λ 0 (A) of finite type. This class contains properly the power series spaces of finite type. Of main interest is its spectrum, which is distinctly different in the cases when λ 0 (A) is nuclear and when it is not. The nuclearity of λ 0 (A) is characterized via certain properties of the spectrum of C. Moreover, C is always power bounded and uniformly mean ergodic on λ 0 (A).

In this study, proximity based optimization algorithms are used for lossy compression of hyperspectral images that are inherently large scale. This is the first time that such proximity based optimization algorithms are implemented with an online dictionary learning method. Compression performances are compared with the one obtained by various sparse representation algorithms. As a result, proximity based optimization algorithms are listed among the three best ones in terms of compression performance values for all hyperspectral images. Additionally, the applicability of anomaly detection is tested on the reconstructed images.

We study some classical operators defined on the weighted Bergman Fréchet space A p α+ (resp. weighted Bergman (LB)-space A p α− ) arising as the projective limit (resp. inductive limit) of the standard weighted Bergman spaces into the growth Fréchet space H ∞ α+ (resp. growth (LB)-space H ∞ α− ), which is the projective limit (resp. inductive limit) of the growth Banach spaces. We show that, for an analytic self map φ of the unit disc D , the continuities of the weighted composition operator Wg,φ , the Volterra integral operator Tg , and the pointwise multiplication operator Mg defined via the identical symbol function are characterized by the same condition determined by the symbol's state of belonging to a Bloch-type space. These results have consequences related to the invertibility of Wg,φ acting on a weighted Bergman Fréchet or (LB)-space. Some results concerning eigenvalues of such composition operators Cφ are presented.

In this paper, a lossy compression of hyperspectral images is realized by using a novel online dictionary learning method in which three dimensional datasets can be compressed. This online dictionary learning method and blind compressive sensing (BCS) algorithm are combined in a hybrid lossy compression framework for the first time in the literature. According to the experimental results, BCS algorithm has the best compression performance when the compression bit rate is higher than or equal to 0.5 bps. Apart from observing rate-distortion performance, anomaly detection performance is also tested on the reconstructed images to measure the information preservation performance.

The spectrum of the Ces?ro operator C is determined on the spaces which arises as intersections Ap ?+ (resp. unions Ap ?-) of Bergman spaces Ap? of order 1 < p < 1 induced by standard radial weights (1-|z|)?, for 0 < ? < 1. We treat them as reduced projective limits (resp. inductive limits) of weighted Bergman spaces Ap?, with respect to ?. Proving that these spaces admit the monomials as a Schauder basis paves the way for using Grothendieck-Pietsch criterion to deduce that we end up with a non-nuclear Fr?chet-Schwartz space (resp. a non-nuclear (DFS)-space). We show that C is always continuous, while it fails to be compact or to have bounded inverse on Ap ?+ and Ap ?-.