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The Calder\'lcurlyborcurlybn Reproducing Formula Associated with the Heisenberg Group
Jinsen Xiao,Jianxun He
Mathematical Problems in Engineering , 2010, DOI: 10.1155/2010/132186
Abstract: We establish the Calderón reproducing formula for functions in 2 on the Heisenberg group H. Also, we develop this formula in (H) with 1<<∞.
Calderón's reproducing formula for Hankel convolution
R. S. Pathak,Gireesh Pandey
International Journal of Mathematics and Mathematical Sciences , 2006, DOI: 10.1155/ijmms/2006/24217
Abstract: Calderón-type reproducing formula for Hankel convolution is established using the theory of Hankel transform.
On boundedness of Calderón-Toeplitz operators  [PDF]
Ondrej Hutník
Mathematics , 2011, DOI: 10.1007/s00020-011-1883-2
Abstract: We study the boundedness of Toeplitz-type operators defined in the context of the Calder\'on reproducing formula considering the specific wavelets whose Fourier transforms are related to Laguerre polynomials. Some sufficient conditions for simultaneous boundedness of these Calder\'on-Toeplitz operators on each wavelet subspace for unbounded symbols are given, where investigating the behavior of certain sequence of iterated integrals of symbols is helpful. A number of examples and counterexamples is given.
Reproducing subgroups of Sp(2,R). Part II: admissible vectors  [PDF]
Giovanni S. Alberti,Filippo De Mari,Ernesto De Vito,Lucia Mantovani
Mathematics , 2012, DOI: 10.1007/s00605-013-0542-x
Abstract: In part I we introduced the class ${\mathcal E}_2$ of Lie subgroups of $Sp(2,\R)$ and obtained a classification up to conjugation (Theorem 1.1). Here, we determine for which of these groups the restriction of the metaplectic representation gives rise to a reproducing formula. In all the positive cases we characterize the admissible vectors with a generalized Calder\'on equation. They include products of 1D-wavelets, directional wavelets, shearlets, and many new examples.
A reproducing kernel for nonsymmetric Macdonald polynomials  [PDF]
K. Mimachi,M. Noumi
Mathematics , 1996,
Abstract: We present a new formula of Cauchy type for the nonsymmetric Macdonald polynomials which are joint eigenfunctions of q-Dunkl operators. This gives the explicit formula for a reproducing kernel on the polynomial ring of n variables.
An APOS analysis of natural science students’ understanding of derivatives
A Maharaj
South African Journal of Education , 2013,
Abstract: This article reports on a study which used the APOS (action-process-object-schema) theoretical framework to investigate university students’ understanding of derivatives and their applica-tions. Research was done at the Westville Campus of the University of KwaZulu-Natal in South Africa. The relevant rules for finding derivatives and their applications were taught to under-graduate science students. This paper reports on the analysis of students’ responses to six types of questions on derivatives and their applications. The findings of this study suggest that those students had difficulty in applying the rules for derivatives and this was possibly the result of many students not having appropriate mental structures at the process, object and schema lev-els.
An APOS analysis of natural science students' understanding of derivatives
Aneshkumar Maharaj
South African Journal of Education , 2013,
Abstract: This article reports on a study which used the APOS (action-process-object-schema) theoretical framework to investigate university students' understanding of derivatives and their applications. Research was done at the Westville Campus of the University of KwaZulu-Natal in South Africa. The relevant rules for finding derivatives and their applications were taught to undergraduate science students. This paper reports on the analysis of students' responses to six types of questions on derivatives and their applications. The findings of this study suggest that those students had difficulty in applying the rules for derivatives and this was possibly the result of many students not having appropriate mental structures at the process, object and schema levels.
The atomic decomposition of harmonic functions satisfying certain conditions of integrability  [cached]
Krzysztof Bogdan
International Journal of Mathematics and Mathematical Sciences , 1995, DOI: 10.1155/s0161171295000809
Abstract: Distributions on Euclidean spaces with derivatives of their Poisson integral satisfying certain natural conditions of integrability are represented as sums of weighted atoms. The atomic decomposition is obtained by means of the Calder 3n reproducing formula.
Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets  [cached]
Mohamed Ali Mourou
Symmetry, Integrability and Geometry : Methods and Applications , 2009,
Abstract: We prove a Calderón reproducing formula for the Dunkl continuous wavelet transform on R. We apply this result to derive new inversion formulas for the dual Dunkl-Sonine integral transform.
Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets  [PDF]
Mohamed Ali Mourou
Mathematics , 2009, DOI: 10.3842/SIGMA.2009.071
Abstract: We prove a Calder\'on reproducing formula for the Dunkl continuous wavelet transform on $\mathbb{R}$. We apply this result to derive new inversion formulas for the dual Dunkl-Sonine integral transform.
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