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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] |
Short-time Fourier Transform
The short-time Fourier transform (STFT) is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In practice, the procedure for computing STFTs is to divide a longer time signal into shorter segments of equal length and then compute the Fourier transform separately on each shorter segment. This reveals the Fourier spectrum on each shorter segment. One then usually plots the changing spectra as a function of time, known as a spectrogram or waterfall plot, such as commonly used in software defined radio (SDR) based spectrum displays. Full bandwidth displays covering the whole range of an SDR commonly use fast Fourier transforms (FFTs) with 2^24 points on desktop computers. Forward STFT Continuous-time STFT Simply, in the continuous-time case, the function to be transformed is multiplied by a window function which is nonzero for only a short period of time. The Fourier transform (a o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Schwartz Space
In mathematics, Schwartz space \mathcal is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space \mathcal^* of \mathcal, that is, for tempered distributions. A function in the Schwartz space is sometimes called a Schwartz function. Schwartz space is named after French mathematician Laurent Schwartz. Definition Let \mathbb be the set of non-negative integers, and for any n \in \mathbb, let \mathbb^n := \underbrace_ be the ''n''-fold Cartesian product. The ''Schwartz space'' or space of rapidly decreasing functions on \mathbb^n is the function space\mathcal \left(\mathbb^n, \mathbb\right) := \left \,where C^(\mathbb^n, \mathbb) is the function space of smooth functions from \mathbb^n into \mathbb, and\, f\, _:= \sup_ \left, \boldsymbol^\boldsymbol (\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hans Georg Feichtinger
Hans Georg Feichtinger (born 16 June 1951) is an Austrian mathematician. He is Professor in the mathematical faculty of the University of Vienna. He is editor-in-chief of the Journal of Fourier Analysis and Applications (JFAA) and associate editor to several other journals. He is one of the founders and head of the Numerical Harmonic Analysis Group (NuHAG) at University of Vienna. Today Feichtinger's main field of research is harmonic analysis with a focus on time-frequency analysis. Biography Hans Georg Feichtinger was born in Wiener Neustadt where he graduated from the Gymnasium (school), Gymnasium and received the Matura "summa cum laude" in 1969. In the same year he started his studies in mathematics and physics. He finished his PhD at the University of Vienna in 1974 under the supervision of Hans J. Reiter, Hans Reiter in 1974 with a doctoral thesis on ''Subalgebras of L1(G)''. Feichtinger attained professorship with the defense of his habilitation thesis on ''Banach convo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
