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By Kenneth Hoffman
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Extra resources for Banach spaces of analytic functions
Example text
1. x/ dx exists for every a interval Œa; b. The proof is left to the reader as an exercise. The converse of this theorem is not necessarily true. R/. This suggests the following definition. 3 (Locally Integrable Functions). x/ dx exists. x/ dx exists for every positive integer n. is sufficient to check that the integral n The proof of this simple fact is left as an exercise. R/ is a subspace of the space of locally integrable functions. 2. The locally integrable functions form a vector space. The absolute value of a locally integrable function is locally integrable.
2009) gave the construction and characterization of all vectorvalued multivariate wavelet packets associated with dilation matrix by means of time–frequency analysis, matrix theory, and operator theory. Wavelet theory has been studied extensively in both theory and applications during the last two decades. This theory has become a promising tool in signal processing, fractals and image processing, and so on, because of their ability to offer good properties like symmetry, certain regularity, continuity, and short support.
Unfortunately, the ridgelet transform is only applicable to objects with global straight line singularities, which are rarely observed in real applications. For example, in image processing, edges are typically curved rather than straight and ridgelets alone cannot yield efficient representations. To overcome inherent limitations of this transform, Candés and Donoho (2003a) developed a new multiscale transform called curvelet transform, which was designed to represent edges and other singularities along curves much more efficiently than traditional 18 1 Brief Historical Introduction transforms.