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Modulation Space
Modulation spaces are a family of Banach spaces defined by the behavior of the short-time Fourier transform with respect to a test function from the Schwartz space. They were originally proposed by Hans Georg Feichtinger and are recognized to be ''the right kind of function spaces'' for time-frequency analysis. ''#Feichtinger's algebra, Feichtinger's algebra'', while originally introduced as a new Segal algebra, is identical to a certain modulation space and has become a widely used space of test functions for time-frequency analysis. Modulation spaces are defined as follows. For 1\leq p,q \leq \infty , a non-negative function m(x,\omega) on \mathbb^ and a test function g \in \mathcal(\mathbb^d) , the modulation space M^_m(\mathbb^d) is defined by : M^_m(\mathbb^d) = \left\. In the above equation, V_gf denotes the short-time Fourier transform of f with respect to g evaluated at (x,\omega) , namely :V_gf(x,\omega)=\int_f(t)\overlinee^dt=\mathcal^_(\overline\hat(\xi+\o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space". Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
