Probability Modeled Optimal K-Frame for Erasures

It has been found that the research of a special kind of frames, named K-frames where K is an operator, is significant in theory and application. In this paper, first of all, in theory, we discuss some properties about the existence and structure of the K-frames and the relationship between operator K and K-frames. And these properties provide a method to construct K-frames in the following text. Next, in application, in order to use K-frames to deal with the data erasures of communication, we establish a probability model. Based on this probability model and the above-mentioned properties of K-frames, we construct a probability uniform Parseval K-frame, such that its canonical dual is the unique probability optimal dual when kKf k \ kf k = c and kK⁺f k \ kf k = a, where both a and c are constants, where K+ is the pseudoinverse for K. We use kf k to represent the norm of f , wheref is any vector in Hilbert space. Moreover, the existence of the probability uniform Parseval K-frame is discussed and then uses it to handle the problem of data erasures. Furthermore, we obtain the conditions for operator K to form the K-frame, which lead to the reduction of restrictions on frame sequences, and make the construction become more flexible compared with traditional frames in coding. Finally, we use three figures obtained by numerical experiments to compare the decoding effects.