Multivariate wavelet frames in various function spaces and their applications

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With the advance of technology, nowadays we are living in a world which is full of digital data sets such as digital images and videos. In order to process, store, and transmit data efficiently, a mathematical model with desirable properties is in high demand to sparsely and economically represent various types of data sets. In fact, modern information technology is built on many advanced mathematical tools coupled with fast computational algorithms. As a multiscale-based representation simulating the procedure that our human eyes use to analyze images, a wavelet or framelet has proved to be an effective computational tool for sparsely representing a wide range of digital data sets. For example, wavelet-based methods are now standard tools for image compression and signal denoising.