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

ISBN-10: 1461463920

ISBN-13: 9781461463924

ISBN-10: 1461463939

ISBN-13: 9781461463931

Advances in utilized arithmetic and Approximation idea: Contributions from AMAT 2012 is a suite of the easiest articles awarded 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 protecting subject matters similar to ODEs, PDEs, distinction equations, utilized research, computational research, sign conception, confident operators, statistical approximation, fuzzy approximation, fractional research, semigroups, inequalities, targeted services and summability. the gathering can be an invaluable source for researchers in utilized arithmetic, engineering and statistics.​

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M, distinct prime numbers. Then b a α (x − a) m ∏ λi ρ Δ a i f i (x) Γ (νi −ρi +1) (x−a)(νi −ρi ) dx ≤ i=1 γ (b − a)α −m+1 α −m+1 m ∏ i=1 a b ν λi Δ a i f i (x) dx . 99) Proof. 51). 31 ([2]). Let ν > 0, n := [ν ], α = ν − n, 0 < α < 1, f ∈ C ([a, b]). Consider ν 1−α (n) Cb− ([a, b]) := { f ∈ Cn ([a, b]) : Ib− f ∈ C1 ([a, b])}. 100) Define the right generalized ν -fractional derivative of f over [a, b], by ν 1−α (n) Δb− f := (−1)n−1 Ib− f . 0 f = f .

M. 41) (x−y)αi −1 Γ (α i ) , a < y ≤ x, 0, x < y < b. 43) a < x < b, i = 1, . . , m. Notice that χ(a,x] (y) (x − y)αi −1 Γ (αi + 1) · Γ (αi ) (x − a)αi m ki (x, y) ∏ Ki (x) = ∏ i=1 i=1 m m m ∏ i=1 αi −1 χ(a,x] (y) (x − y) (x − a)αi αi = χ(a,x] (y) (x − y) ∑ αi −m i=1 m ∏ αi i=1 m (x − a) = ∑ αi . 45) α −m γ ki (x, y) χ(a,x] (y) (x − y) = . 32), we get for appropriate weight u that λm (y) = γ for all a < y < b. 47) 2 Univariate Hardy-Type Fractional Inequalities 33 Let Φi : R+ → R+ , i = 1, .

Here Γ (α ) is the Gamma function. These integrals are called the leftsided and the right-sided fractional integrals. We mention some properties of the α f and I α f of order α > 0; see also [13]. 3) where K= (b − a)α . G. Hardy in one of his first papers; see [10]. He did not write down the constant, but the calculation of the constant was hidden inside his proof. Next we follow [11]. Let (Ω1 , Σ1 , μ1 ) and (Ω2 , Σ2 , μ2 ) be measure spaces with positive σ -finite measures, and let k : Ω1 × Ω2 → R be a nonnegative measurable function, k (x, ·) measurable on Ω2 and K (x) = Ω2 k (x, y) d μ2 (y) , x ∈ Ω1 .

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Advances in Applied Mathematics and Approximation Theory: Contributions from AMAT 2012 by George A. Anastassiou (auth.), George A. Anastassiou, Oktay Duman (eds.)

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