We will introduce the Euler-Maruyama and Milstein Methods. By using pseudo-random number and implementing the Ito integral we will present an efficient algorithm for the PATHWISE numerical APPROXIMATION of solutions to stochastic differential equations.In the last section we will show the efficiency of the algorithm by presenting the numerical results for some test problems.

This paper seeks to prove the PATHWISE uniqueness of an abstract stochastic partial differential equation in Hilbert spaces driven by both Poisson random measure and the Wiener process with Hölder continuous drift. The main idea is based on the corresponding infinite-dimensional Kolmogorov equation. In addition, the main result is further supported by the help of an example.

‎‎‎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.

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.