Study Of Some Questions For Continuous Frames

The concept of frame was firstly introduced by Dufn and Schaefer when they studiednonharmonic Fourier series. A discrete frame in a separable Hilbert space is almost asgood as an orthonormal basis for expanding arbitrary elements. Continuous frames, whichare extensive and general frames, were firstly studied by Ali, Antoine and Gazeau in1993.The elements of the continuous frames are a family of vectors from a Hilbert space Hindexed by X, which is a locally compact space with measure. The discrete frames arespecial forms of continuous frames. The conceptions of g-frames and g-Riesz bases weregiven by Sun in2006. And then in2008, Dehghan and Hasankhani Fard studied somequalities of g-continuous frames. Afterwards, Antoine and Balazs introduced the notionof semi-frames in2011. We speak of an semi-frame if only one of its frame bounds isvalid. Frame also has many extensive application in signal processing, image processing,numerical calculation and many other disciplines. Based on the theory of discrete framesand g-frames, this thesis is mainly to study the qualities and perturbation of continuousframes. Meanwhile it studies some qualities and determination methods of continuoussemi-frames, continuous Bessel sequences and continuous frame sequences. At last, somesimple qualities of g-semi-frames are studied. It consists of five chapters.Chapter1states the origin of frames, continuous fames and semi-frames. And thenit sketches the structure and works of the thesis.Chapter2lists some basic theorems to be used throughout the thesis, the conceptsof discrete frames, continuous frames, semi-frames, g-frames and g-continuous frames.Chapter3is one of the main contents. It studies some determination methods ofcontinuous semi-frames and continuous Bessel sequences.Chapter4is also the main content of the thesis. It promotes some good qualities ofdiscrete frames to continuous frames. Then it studies some determination methods andperturbation of continuous frames.Lastly, it studies some simple qualities of g-semi-frames.