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

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

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Lm2 (Lm1 f )) (x) − f (x) ≤ Lmr Lmr−1 (. . 37) r ≤ ∑ Lmi f − f ∞ i=1 r c N ∑ ω1 f, ω1 f, i=1 rcN 1 β mi 1 β m1 +μ f ∞e +μ f ∞e (1−β ) −γ mi (1−β ) −γ m1 ∞ ≤ ≤ . Clearly, we notice that the speed of convergence of the multiply iterated operator equals to the speed of Lm1 . Proof. We write Lmr Lmr−1 (. . Lm2 (Lm1 f )) − f = Lmr Lmr−1 (. . Lm2 (Lm1 f )) − Lmr Lmr−1 (. . Lm2 f ) + Lmr Lmr−1 (. . Lm2 f ) − Lmr Lmr−1 . . Lm3 f Lmr Lmr−1 . . Lm3 f + − Lmr Lmr−1 (. . Lm4 f ) + . + Lmr Lmr−1 f − Lmr f + Lmr f − f = Lmr Lmr−1 (.

72) i=1 αi where α > m − 1, fi with Ib− (| fi |) finite and fi Lebesgue integrable, i = 1, . . , m. Next let pi > 1, and Φi (x) = x pi , x ∈ R+ . 68), we get m b I2 := (b − x)α a ∏ i=1 γ ∏ (Γ (αi + 1)) i=1 b ⎜ ⎜ ⎝ a pi (α − m + 1) b i=1 a i= j pj ⎟ | fi (x)| pi dx⎠ · dx ≤ ⎞ γ (b − a) ∏ (Γ (αi + 1)) ⎞ ⎟⎜ m ⎟ ⎝∏ ⎠ (x − a)α −m+1 f j (x) α −m+1 m dx ≤ m (b − x)i=1 ⎞⎛ m ⎛ pi ∑ αi pi ⎛ ⎜ ⎜ ⎝ αi Ib− fi (x) pi i=1 (α − m + 1) ⎟ ⎟ ⎠ m ∏ i=1 a b | fi (x)| pi dx . 73) Notice here that β := α − ∑ αi pi < 0.

E. in x ∈ Ω1 . 6). 1 of [11]. 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 . Define λ2 on Ω2 by λ2 (y) := Ω1 u(x)k1 (x,y)k2 (x,y) K1 (x)K2 (x) is integrable on Ω1 , u (x) k1 (x, y) k2 (x, y) d μ1 (x) < ∞.