Mathematical Researches

fa مجموعه‌های موجک روی گروه‌های آبلی موضعاً فشرده Wavelet Sets on Locally Compact Abelian Groups جبر alg مقاله استخراج شده از طرح پژوهشی Research Paper در این مقاله مفهوم مجموعه‌های موجک روی گروه‌های آبلی موضعاً فشرده با شبکه یکنواخت تعریف شده است. این تعریف تعمیمی از مجموعه‌های موجک در فضای اقلیدسی است. سپس با استفاده از تبدیل فوریه و تجزیۀ چند ریزه‌ساز این مجموعه‌ها مشخص شده‌اند. در ادامه مجموعه‌های مقیاس تعمیم یافته روی گروه‌های آبلی موضعاً فشرده تعریف و بررسی شده‌اند و ارتباط بین مجموعه‌های مقیاس تعمیم یافته و مجموعه‌های موجک بیان شده است. در پایان با تعریف تابع بعد موجک روی گروه‌های آبلی موضعاً فشرده، مجموعه‌های مقیاس تعمیم یافته مشخص شده  و رابطۀ آنها با مجموعه‌های موجک بررسی شده است. Introduction An orthonormal wavelet is a square-integrable function whose translates and dilates form an orthonormal basis for the Hilbert space <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="37" >. That is, given the unitary operators of translation <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="121" > for <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="35" > and dilation <img alt="" height="21" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="113" >, we call <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="65" > an orthonormal wavelet if the set <img alt="" height="27" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="300" >is an orthonormal basis for <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="37" >. This definition was later generalized to higher dimensions and to allow for other dilation and translation sets; let Hilbert space <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image014.gif" width="45" > and an n × n expansive matrix A (i.e. a matrix with eigenvalues bigger than 1) with integer entries, then dilation operator <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="16" > is given by <img alt="" height="23" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="132" > and the translation operator <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image020.gif" width="14" > is given by <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="121" > for <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image022.gif" width="43" >. A finite set <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image024.gif" width="202" > is called multiwavelet if the set <img alt="" height="23" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image026.gif" width="216" >, is an orthogonal basis for <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image014.gif" width="45" >. The concept of a multiresolution analysis, abbreviated as MRA is Central to the theory of wavelets. There is much overlaps between wavelet analysis and Fourier analysis. Indeed, wavelets can be thought of as non-trigonometric Fourier series. Thus, Fourier analysis is used as a tool to investigate properties of wavelets. Another concept is wavelet set. The term wavelet set was coined by Dai and Larson in the late 90s to describe a set W such that <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image028.gif" width="20" >, the characteristic function of W, is the Fourier transform of an orthonormal wavelet on <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="37" >. At about the same time as the Dai and Larson paper, Fang and Wang first used the term MSF wavelet (minimally supported frequency wavelet) to describe wavelets whose Fourier transforms are supported on sets of the smallest possible measure. The importance of MSF wavelets as a source of examples and counterexamples has continued throughout wavelet history. A famous example due to Journe first showed that not all wavelets have an associated structure multiresolution analysis (MRA). The discovery of a non-MRA wavelet gave an important push to the development of more general structures such as frame multiresolution analyses (FMRAs) and generalized multiresolution analysis(GMRAs). In this paper we generalize wavelets and wavelet sets on locally compact Abelian group G with uniform lattice.   Material and methods In this paper, we investigate wavelet sets on locally compact abelian groups with uniform lattice, where a uniform lattice H in LCA group G is a discrete subgroup of G such that the quotient group G/H is compact.  So we review some basic facts from the theory of LCA groups and harmonic analysis. Then we define wavelet sets on these groups and characterize them by using of Fourier transform and multiresolution analysis. Results and discussion We extend theory of wavelet sets on locally compact abelian groups with uniform lattice. This is a generalization of wavelet sets on Euclidean space. We characterize wavelet sets by using of Fourier transform and multiresolution analysis. Also, we define generalized scaling sets and dimension functions on locally compact abelian groups and verify its relations with wavelet sets. Dimension functions for MSF wavelets are described by generalized scaling sets. In the setting of LCA groups, we define translation congruent and show wavelet sets are translation congruent, so we can define a map on G such that it is measurable, measure preserving and bijection. Conclusion The following conclusions were drawn from this research. <ul> <li>Wavelet sets on locally compact groups by uniform lattice can be defined. This is a generalization of wavelet sets on Euclidean space.</li> <li>Characterization of wavelet sets on LCA groups can be done in different ways. A method is to use Fourier transform and translation congruent. Another way is to generaliz scaling set and dimension function.</li> <li>As an example, Cantor dyadic group is a non-trivial example that satisfies in the theory of wavelet sets on locally compact groups by uniform lattice. We find wavelet set and generalized scaling set for this group and show related wavelet is MRA wavelet.<a href="./files/site1/files/63/7.pdf">./files/site1/files/63/7.pdf</a></li> </ul> موجک, گروه آبلی موضعاً فشرده, مجموعه موجک, تجزیۀ چند ریزه‌ساز, مجموعه‌های مقیاس تعمیم یافته. wavelet, locally compact abelian group, wavelet set, multiresolution analysis, generalized scaling sets. 393 404 http://mmr.khu.ac.ir/browse.php?a_code=A-10-196-1&slc_lang=fa&sid=1 Mehdi Rashidi-Kouchi مهدی رشیدی کوچی rashidiehdi20@gail.com 10031947532846004032 10031947532846004032 Yes دانشگاه آزاد اسلامی، واحد کهنوج، گروه ریاضی