By George A. Anastassiou (auth.), George A. Anastassiou, Oktay Duman (eds.)
Advances in utilized arithmetic and Approximation thought: Contributions from AMAT 2012 is a set of the simplest articles offered at “Applied arithmetic and Approximation thought 2012,” a global convention held in Ankara, Turkey, may possibly 17-20, 2012. This quantity brings jointly key paintings from authors within the box masking issues reminiscent of ODEs, PDEs, distinction equations, utilized research, computational research, sign thought, optimistic operators, statistical approximation, fuzzy approximation, fractional research, semigroups, inequalities, particular services and summability. the gathering might be an invaluable source for researchers in utilized arithmetic, engineering and statistics.
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Extra resources for Advances in Applied Mathematics and Approximation Theory: Contributions from AMAT 2012
The rest of the summands of Gn ( f , x1 , x2 ) are treated all the same way and similarly to the case of N = 1. The method is demonstrated as follows. −λ2 k1 k2 We will prove that ∑∞ k1 =λ1 ∑k2 =−∞ f n , n Φ1 (nx1 − k1 ) Φ2 (nx2 − k2 ) is continuous in (x1 , x2 ) ∈ R2 . The continuous function k1 k2 , n n f Φ1 (nx1 − k1) Φ2 (nx2 − k2 ) ≤ f and −λ2 ∞ f ∞ ∑ ∑ k1 =λ1 k2 =−∞ ∞ f ∑ ∞ k1 =λ1 ⎛ f ∞ Φ1 (k1 − λ1 ) Φ2 (k2 + λ2 ) = −λ2 ∑ Φ1 (k1 − λ1) ∞ k2 =−∞ ⎞⎛ ⎝ ∑ Φ1 k1 ⎠ ⎝ k1 =0 ∞ Φ1 (k1 − λ1 ) Φ2 (k2 + λ2 ) , Φ2 (k2 + λ2) ≤ ⎞ 0 ∑ Φ2 k2 ⎠ ≤ f ∞.
Define ν on Ω2 by K(x) ν (y) := Ω1 u (x) k (x, y) d μ1 (x) < ∞. e. in x ∈ Ω1 . 6). 1 of . 1 for products of several functions and we give wide applications to fractional calculus. 9) i = 1, . . , m. e. on Ω1 and the weight functions are nonnegative measurable functions on the related set. 10) where fi : Ω2 → R are measurable functions, i = 1, . . , m. Here u stands for a weight function on Ω1 . The first introductory result is proved for m = 2. 2. Assume that the function x → for each y ∈ Ω2 .
3984)e−m1 . 47) One can give easily many other interesting applications. Acknowledgement The author wishes to thank his colleague Professor Robert Kozma for introducing him to the general theory of cellular neural networks that motivated him to consider and study iterated neural networks. References 1. A. Anastassiou, Rate of convergence of some neural network operators to the unitunivariate case, J. Math. Anal. Appli. 212 (1997), 237–262. 2. A. Anastassiou, Rate of convergence of some multivariate neural network operators to the unit, J.