By George A. Anastassiou (auth.), George A. Anastassiou, Oktay Duman (eds.)

Advances in utilized arithmetic and Approximation conception: Contributions from AMAT 2012 is a set of the simplest articles provided at “Applied arithmetic and Approximation thought 2012,” a global convention held in Ankara, Turkey, could 17-20, 2012. This quantity brings jointly key paintings from authors within the box protecting subject matters reminiscent of ODEs, PDEs, distinction equations, utilized research, computational research, sign idea, optimistic operators, statistical approximation, fuzzy approximation, fractional research, semigroups, inequalities, distinctive services and summability. the gathering could be an invaluable source for researchers in utilized arithmetic, engineering and statistics.

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**Example text**

Define λm on Ω2 by i=1 ⎛ λm (y) := ⎞ m ∏ ki (x, y) ⎟ ⎜ u (x) i=1 ⎜ ⎟ d μ1 (x) < ∞. 38) λm (y) d μ2 (y) , true for all measurable functions, i = 1, . . e. in x ∈ Ω1 . (ii) λm f j α α ; | f1 |α , | f2 |α , | f3 |α , . . , f j , . . 10). Above f j item. 11. Let fi be Lebesgue measurable functions from (a, b) into R, such that αi Ia+ (| fi |) (x) ∈ R, ∀ x ∈ (a, b), αi > 0, i = 1, . . , when fi ∈ L∞ (a, b). 39) fi (x) , x ∈ (a, b), i = 1, . . , m, gi (x) = Ia+ we remind αi Ia+ fi (x) = 1 Γ (αi ) x a (x − t)αi −1 fi (t) dt.

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.

Proof. Here we use Jensen’s inequality, Fubini’s theorem, and that Φi are increasing. 19) 26 George A. 24) 28 George A. 25) proving the claim. For general m ∈ N, the following result is valid. 4. Assume that the function x → ⎝ m u(x) ∏ ki (x,y) i=1 m ∏ Ki (x) ⎞ ⎠ is integrable on Ω1 , i=1 for each y ∈ Ω2 . Define λm on Ω2 by ⎞ ⎛ m k (x, y) u (x) ∏ i ⎟ ⎜ i=1 ⎟ d μ1 (x) < ∞. 26) i=1 Here Φi : R+ → R+ , i = 1, . . , m, are convex and increasing functions. 27) Φ1 (| f1 (y)|) λm (y) d μ2 (y) , true for all measurable functions, i = 1, .