Abstract
Given s G R and 1 < p, q < oo the modulation space Mp g (R m ) can be described as follows, using the Gauss-function go,go(x) '•= exp(-x 2 )for the modulation operator.Among these spaces one has the classical potential spaces £2( R m) _ jv/| 2 (R m ) and the remarkable Segal algebra 5 0 (R m ) = Mf^R™).It is the aim of this paper to show that for these spaces an atomic characterization similar to known characterization of Besov spaces can be given (with dilation being replaced by modulation).Our main theorem is the following: Given s G R and some So # 0, go G Mj*j(R m ) (e.g., g G 5(R m ) or g G L 1 with compactly supported Fourier transform) one has:THEOREM .There exist ao > 0 and ßo > 0 such that, for a < a 0 and ß < ßo, there exists C = C(a,ß) > 0 with the following property: f e M^q(R m ) if and only iff = Yin,k a n,kMß n L ak go, for some double sequence of coefficients satisfying [E (El 0 ».*!")*^1 + Wr] 1/9 < c||/|M p yR m )||. n kThe convergence is in the sense of tempered distributions, and in the norm sense for p,q < oo.
Keywords
Modulation spaceMathematicsType (biology)Modulation (music)Space (punctuation)Pure mathematicsGaussFunction spaceMathematical analysisTopology (electrical circuits)CombinatoricsQuantum mechanicsPhysics
Related Publications
Optimal Global Rates of Convergence for Nonparametric Regression
Consider a $p$-times differentiable unknown regression function $\\theta$ of a $d$-dimensional measurement variable. Let $T(\\theta)$ denote a derivative of $\\theta$ of order $...
Quantum-Mechanically Correct Form of Hamiltonian Function for Conservative Systems
Dirac showed that, if in the Hamiltonian $H$ momenta ${\ensuremath{\eta}}_{r}$ conjugate to the co-ordinates ${\ensuremath{\xi}}_{r}$ are replaced by $(\frac{h}{2\ensuremath{\pi...
Empirical graph Laplacian approximation of Laplace–Beltrami operators: Large sample results
Let ${M}$ be a compact Riemannian submanifold of ${{\\bf R}^m}$ of dimension\n$\\scriptstyle{d}$ and let ${X_1,...,X_n}$ be a sample of i.i.d. points in ${M}$\nwith uniform dist...
Random Fourier transforms
Paley and Zygmund [10](1) have shown that for almost every random choice of signs the series represents a continuous function, provided only that X fln(lg n) < 00 .A little late...
Spin scaling of the electron-gas correlation energy in the high-density limit
The ground-state correlation energy per particle in a uniform electron gas with spin densities ${\mathit{n}}_{\mathrm{\ensuremath{\uparrow}}}$ and ${\mathit{n}}_{\mathrm{\ensure...
Publication Info
- Year
- 1989
- Type
- article
- Volume
- 19
- Issue
- 1
- Citations
- 141
- Access
- Closed
External Links
Social Impact
Altmetric
PlumX Metrics
Social media, news, blog, policy document mentions
Citation Metrics
141
OpenAlex
Cite This
Hans G. Feichtinger
(1989).
Atomic characterizations of modulation spaces through Gabor-type representations.
Rocky Mountain Journal of Mathematics
, 19
(1)
.
https://doi.org/10.1216/rmj-1989-19-1-113
Identifiers
- DOI
- 10.1216/rmj-1989-19-1-113