Bessel Sequences and Its F-Scalability (Advances in Applied Mathematics and Mechanics, 2018)
Frame theory, which contains wavelet analysis and Gabor analysis, has becomea powerful tool for many applications of mathematics, engineering and quantummechanics. The study of extension principles of Bessel sequences to frames is importantin frame theory. This paper studies transformations on Bessel sequences to generateframes and Riesz bases in terms of operators and scalability. Some characterizationsof operators that mapping Bessel sequences to frames and Riesz bases are given. Weintroduce the definitions of F-scalable and P-scalable Bessel sequences. F-scalabilityand P-scalability of Bessel sequences are discussed in this paper, then characterizationsof scalings of F-scalable or P-scalable Bessel sequences are established. Finally, aperturbation result on F-scalable Bessel sequences is derived.