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Let X be a Banach space and let Y be a separable Lindenstrauss space. We describe the Banach space P (Y, X) of absolutely summing operators as a general L1-tree space. We also characterize the bounded APPROXIMATION PROPERTY and its weak version for X in terms of the space of integral operators I (X, Z*) and the space of nuclear operators N (X, Z*), respectively, where Z is a Lindenstrauss space, whose dual Z* fails to have the Radon-Nikod´ym PROPERTY.

‎‎‎Given a fuzzy normed space, we will introduce the notion of fuzzy near best APPROXIMATION as a generalization of the notion of fuzzy best APPROXIMATION. Some basic properties are characterized and also many examples for illustration are presented. Also, the hereditary properties of the fuzzy near best APPROXIMATION on direct sum and tensor product of linear spaces are discussed.

The previous part of this essay discussed the decree for possessing a found PROPERTY. Here in this part, the author reviews three topics: giving charity, safekeeping a PROPERTY, and handing over a found PROPERTY to the legal ruler. To elaborate each topic, he discusses evidences mentioned in Quran and Tradition and examines their authenticity in comparison to the ones of derelict PROPERTY. At the end, the decrees of a derelict PROPERTY and the fact of handing it over to the legal ruler are discussed.

The essay focuses on taking possession of the "Found PROPERTY". The author first presents four types of narrations that indicate taking possession of the Found PROPERTY, and then discusses their possible interpretations. He, then, mentions three ways to solve the paradoxes that lie in those narrations, and reviews the cases in which one can or should give the Found PROPERTY as charity. He believes the Found Animal and the Found PROPERTY in Masjid -Al- Haram are exceptions and one cannot take possession of them. Finally, he finds the relation between narrations which suggest taking possession of the Found PROPERTY or giving it as charity and those which highlight that some Found Properties cannot be possessed.

Over the last decade or so, wavelets have had a growing impact on signal processing theory and practice, both because of their unifying role and their successes in applications. Filter banks, which lie at the heart of wavelet-based algorithms, have become standard signal processing operators, used routinely in applications ranging from compression to modems. The contributions of wavelets have often been in the subtle interplay between discrete-time and continuous-time signal processing. The purpose of this article is to look at recent wavelet advances from a signal processing perspective. In particular, APPROXIMATION results are reviewed, and the implication on compression algorithms is discussed. New constructions and open problems are also addressed. Finding a good basis to solve aproblem dates at least back to Fourier and his investigation of the heat equation. The series proposed by Fourier has several distinguishing features: The series is able to represent any finite energy function on an interval. The basis functions are eigenfunctions of linear shift invariant systems or, in other words, Fourier series diagonalize linear, shift invariant operators.

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In this paper, we study wavelet APPROXIMATION of the Chebyshev polyno-mials of the first, second, third, and fourth kinds. We estimate the wavelet APPROXIMATION of a function f having bounded first derivatives.