كليدواژه :
فضاي هيلبرت , دنبالۀ بسل , قاب , دوگان تقريبي , ضرب گر , بازسازي سيگنال
چكيده فارسي :
در اين مقاله انواع جديدي از دوگان ها و دوگان هاي تقريبي در فضاهاي هيلبرت را با استفاده از ضرب گرها، عملگرهاي وارون پذير و نشانه ها معرفي مي‌كنيم. تاكنون مقالات متعددي در مورد دوگان هاي تقريبي و كاربردهاي آنها نوشته شده كه در اين مقالات دوگان هاي تقريبي براي دنباله هاي بسل بررسي شده‌اند. در اين‌جا دوگان هاي تقريبي را براي دنباله هاي دلخواه در يك فضاي هيلبرت تعريف كرده، آنها را با دوگان هاي تقريبي بسل مورد مقايسه قرارداده و نشان مي دهيم با وجود اينكه اين دوگان هاي تقريبي لزوماً تمام خواص دوگان هاي تقريبي بسل را ندارند اما مي توانند در بازسازي سيگنال ها مفيد واقع شوند. علاوه براين، نتايج جديدي براي دوگان هاي تقريبي بسل به دست مي آوريم.
چكيده لاتين :
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.
