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A reproducing system is a countable collection of functions such that a general function f can be decomposed as , with some control on the analyzing coefficients . Several such systems have been introduced very successfully in mathematics and its applications. We present a unified viewpoint in the study of reproducing systems on locally compact abelian groups G. This approach gives a novel characterization of the Parseval frame generators for a very general class of reproducing systems on L²(G). As an application, we obtain a new characterization of Parseval frame generators for Gabor and affine systems on L²(G).
Gitta Kutyniok, and Demetrio Labate. "The theory of reproducing systems on locally compact abelian groups." Colloquium Mathematicae 106.2 (2006): 197-220. <http://eudml.org/doc/286281>.
@article{GittaKutyniok2006, abstract = {A reproducing system is a countable collection of functions $\{ϕ_\{j\}: j ∈ \}$ such that a general function f can be decomposed as $f = ∑_\{j∈\} c_\{j\}(f)ϕ_\{j\}$, with some control on the analyzing coefficients $c_\{j\}(f)$. Several such systems have been introduced very successfully in mathematics and its applications. We present a unified viewpoint in the study of reproducing systems on locally compact abelian groups G. This approach gives a novel characterization of the Parseval frame generators for a very general class of reproducing systems on L²(G). As an application, we obtain a new characterization of Parseval frame generators for Gabor and affine systems on L²(G).}, author = {Gitta Kutyniok, Demetrio Labate}, journal = {Colloquium Mathematicae}, keywords = {locally compact Abelian group; duality; Plancherel formula; frames; wavelets; Gabor systems; affine systems}, language = {eng}, number = {2}, pages = {197-220}, title = {The theory of reproducing systems on locally compact abelian groups}, url = {http://eudml.org/doc/286281}, volume = {106}, year = {2006}, }
TY - JOUR AU - Gitta Kutyniok AU - Demetrio Labate TI - The theory of reproducing systems on locally compact abelian groups JO - Colloquium Mathematicae PY - 2006 VL - 106 IS - 2 SP - 197 EP - 220 AB - A reproducing system is a countable collection of functions ${ϕ_{j}: j ∈ }$ such that a general function f can be decomposed as $f = ∑_{j∈} c_{j}(f)ϕ_{j}$, with some control on the analyzing coefficients $c_{j}(f)$. Several such systems have been introduced very successfully in mathematics and its applications. We present a unified viewpoint in the study of reproducing systems on locally compact abelian groups G. This approach gives a novel characterization of the Parseval frame generators for a very general class of reproducing systems on L²(G). As an application, we obtain a new characterization of Parseval frame generators for Gabor and affine systems on L²(G). LA - eng KW - locally compact Abelian group; duality; Plancherel formula; frames; wavelets; Gabor systems; affine systems UR - http://eudml.org/doc/286281 ER -