Multivariate wavelet frames in various function spaces and their applications

随着科技的进步，如今我们生活在一个充满数字数据集的世界，如数字图像和视频。为了高效地处理、存储和传输数据，需要一种具有理想性质的数学模型来稀疏而经济地表示各种类型的数据集。事实上，现代信息技术是建立在许多先进的数学工具和快速计算算法的基础上的。作为一种模拟人眼分析图像过程的多尺度表示，小波或小框架已被证明是一种有效的计算工具，可用于稀疏表示广泛的数字数据集。例如，基于小波的方法现在是图像压缩和信号去噪的标准工具。

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.