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پژوهش های ریاضی، جلد ۵، شماره ۲، صفحات ۲۳۷-۲۵۲
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| عنوان فارسی |
دوگان های تقریبی تعریف شده به‌وسیلۀ ضرب گرها و نقش آن ها در بازسازی سیگنال ها |
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| چکیده فارسی مقاله |
در این مقاله انواع جدیدی از دوگانها و دوگانهای تقریبی در فضاهای هیلبرت را با استفاده از ضربگرها، عملگرهای وارونپذیر و نشانهها معرفی می‌کنیم. تاکنون مقالات متعددی در مورد دوگانهای تقریبی و کاربردهای آنها نوشته شده که در این مقالات دوگانهای تقریبی برای دنبالههای بسل بررسی شده‌اند. در این‌جا دوگانهای تقریبی را برای دنبالههای دلخواه در یک فضای هیلبرت تعریف کرده، آنها را با دوگانهای تقریبی بسل مورد مقایسه قرارداده و نشان میدهیم با وجود اینکه این دوگانهای تقریبی لزوماً تمام خواص دوگانهای تقریبی بسل را ندارند اما میتوانند در بازسازی سیگنالها مفید واقع شوند. علاوه براین، نتایج جدیدی برای دوگانهای تقریبی بسل بهدست میآوریم.
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| کلیدواژه‌های فارسی مقاله |
فضای هیلبرت، دنبالۀ بسل، قاب، دوگان تقریبی، ضرب‌گر، بازسازی سیگنال. |
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| عنوان انگلیسی |
Approximate Duals Introduced by Multipliers and Their Role in the Reconstruction of Signals |
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| چکیده انگلیسی مقاله |
Paper pages (237-252)
Introduction
Frames for Hilbert spaces were first introduced by Duffin and Schaeffer in 1952 to study some problems in nonharmonic Fourier series, reintroduced in 1986 by Daubechies, Grossmann and Meyer. Various generalizations of frames have been introduced and many applications of them in different branches have been presented.
Bessel multipliers in Hilbert spaces were introduced by Peter Balazs. As we know in frame theory, the composition of the synthesis and analysis operators of a frame is called the frame operator. A multiplier for two Bessel sequences is an operator that combines the analysis operator, a multiplication pattern with a fixed sequence, called the symbol, and the synthesis operator. Bessel multipliers have useful applications, for example they are used for solving approximation problems and they have applications as time-variant filters in acoustical signal processing. We mention that many generalizations of Bessel multipliers have been introduced, also multipliers have been studied for non-Bessel sequences.
Approximate duals in frame theory have important applications, especially are used for the reconstruction of signals when it is difficult to find alternate duals. Approximate duals are useful for wavelets, Gabor systems and in sensor modeling. Approximate duality of frames in Hilbert spaces was recently investigated by Christensen and Laugesen and some interesting applications of approximate duals were obtained. For example, it was shown that how approximate duals can be obtained via perturbation theory and some applications of approximate duals to Gabor frames especially Gabor frames generated by the Gaussian were presented. Afterwards, many authors studied approximate duals of Bessel sequences and many properties and generalizations of them were presented. In this note, we consider approximate duals for arbitrary sequences.
Results and discussion
In this paper, we introduce some new kinds of duals and approximate duals in Hilbert spaces using multipliers, invertible operators and symbols. Many papers about approximate duals and their applications have been written so far which in these papers approximate duals have been considered for Bessel sequences. Here, we introduce approximate duals for arbitrary sequences in a Hilbert space, compare them with Bessel approximate duals and we show that they can be useful for the reconstruction of signals though they do not have all of the properties of Bessel approximate duals. Moreover, we obtain some new results for Bessel approximate duals.
Conclusion
The following conclusions were drawn from this research. - New kinds of duals and approximate duals for arbitrary sequences are introduced using multipliers, invertible operators and symbols.
- Duals and approximate duals of non-Bessel sequences are compared with the Bessel ones and some differences between them are shown by presenting various examples.
- Some properties and applications of duals and approximate duals of non-Bessel sequences are stated.
- Some new results about duals and approximate duals of Bessel sequences are obtained especially some important concepts such as closeness of Bessel sequences, nearly Parseval frames and multipliers with constant symbols are related to approximate duals of frames. ./files/site1/files/52/12.pdf
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| کلیدواژه‌های انگلیسی مقاله |
Hilbert space, Bessel sequence, Frame, Approximate dual, Multiplier, Reconstruction of signals. |
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| نویسندگان مقاله |
مرتضی میرزایی ازندریانی | Morteza Mirzaee Azandaryani
Qom, University of Qom
دانشگاه قم، دانشکدۀ علوم پایه، گروه ریاضی
مهدی رحیمی | Mehdi Rahimi
Qom, University of Qom
دانشگاه قم، دانشکدۀ علوم پایه، گروه ریاضی
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| نشانی اینترنتی |
http://mmr.khu.ac.ir/browse.php?a_code=A-10-233-1&slc_lang=fa&sid=1 |
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| زبان مقاله منتشر شده |
fa |
| موضوعات مقاله منتشر شده |
جبر |
| نوع مقاله منتشر شده |
علمی پژوهشی کاربردی |
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