Modulation space

In a modulation space , the “size” of a function is determined based on its spectrogram. The spectrogram is clearly divided into sections of equal size, the size of which is in turn determined on the basis of their spectrograms; with a similar description of the Besov spaces , the size of these sections increases exponentially. Modulation spaces are a family of Banach spaces in which a function is measured by means of its short-term Fourier transform with a test function in a Schwartz space . Originally examined by Hans Georg Feichtinger , these rooms turned out to be a useful framework for time-frequency analysis .

definition

For a non-negative function on and a test function is the modulation space by ${\displaystyle 1\leq p,q\leq \infty }$ ${\displaystyle m(x,\omega )}$ ${\displaystyle \mathbb {R} ^{2d}}$ ${\displaystyle g\in {\mathcal {S}}(\mathbb {R} ^{d})}$ ${\displaystyle M_{m}^{p,q}(\mathbb {R} ^{d})}$

${\displaystyle M_{m}^{p,q}(\mathbb {R} ^{d})=\left\{f\in {\mathcal {S}}'(\mathbb {R} ^{d})\ :\ \left(\int _{\mathbb {R} ^{d}}\left(\int _{\mathbb {R} ^{d}}|V_{g}f(x,\omega )|^{p}m(x,\omega )^{p}dx\right)^{q/p}d\omega \right)^{1/q}<\infty \right\}.}$

Are defined.

Here is the short-time Fourier transform in terms of when evaluated. That is, is equivalent to . The space does not depend on . The canonical choice for the test function is the Gaussian function . ${\displaystyle V_{g}f}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle (x,\omega )}$ ${\displaystyle f\in M_{m}^{p,q}(\mathbb {R} ^{d})}$ ${\displaystyle V_{g}f\in L_{m}^{p,q}(\mathbb {R} ^{2d})}$ ${\displaystyle M_{m}^{p,q}(\mathbb {R} ^{d})}$ ${\displaystyle g\in {\mathcal {S}}(\mathbb {R} ^{d})}$

Feichtinger algebra

The modulation space with and , therefore, is also called Feichtinger algebra and was designed by Feichtinger originally named because it is the smallest Segal is algebra, which is invariant under time-frequency shifts, so combined translational and modulation operators. is a Banach space embedded in and invariant under the Fourier transform. For this and other reasons, there is an obvious room for test functions in time-frequency analysis. ${\displaystyle p=q=1}$ ${\displaystyle m(x,\omega )=1}$ ${\displaystyle M_{m}^{1,1}(\mathbb {R} ^{d})=M^{1}(\mathbb {R} ^{d})}$ ${\displaystyle S_{0}}$ ${\displaystyle M^{1}(\mathbb {R} ^{d})}$ ${\displaystyle L^{1}(\mathbb {R} ^{d})\cap C_{0}(\mathbb {R} ^{d})}$ ${\displaystyle M^{1}(\mathbb {R} ^{d})}$

Individual evidence

1. ↑ Karlheinz Gröchenig: Foundations of Time-Frequency Analysis. Birkhäuser, Boston 2001, ISBN 978-0817640224
2. ↑ Modulation Spaces: Looking Back and Ahead
3. ^ H. Feichtinger: On a new Segal algebra. Monthly Math. 92, pp. 269–289, 1981, ( online ( memento of the original from September 25, 2006 in the Internet Archive ) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. ).