(untitled) (Advances in Applied Mathematics and Mechanics, 2018)
Frame theory, which contains wavelet analysis and Gabor analysis, has become a powerful tool for many applications of mathematics, engineering and quantum mechanics. The study of extension principles of Bessel sequences to frames is important in frame theory. This paper studies transformations on Bessel sequences to generate frames and Riesz bases in terms of operators and scalability. Some characterizations of operators that mapping Bessel sequences to frames and Riesz bases are given. We introduce the definitions of F-scalable and P-scalable Bessel sequences. F-scalability and P-scalability of Bessel sequences are discussed in this paper, then characterizations of scalings of F-scalable or P-scalable Bessel sequences are established. Finally, a perturbation result on F-scalable Bessel sequences is derived.