By Aleksandar Ivić (auth.), Walter Gautschi, Giuseppe Mastroianni, Themistocles M. Rassias (eds.)

ISBN-10: 1441965939

ISBN-13: 9781441965936

ISBN-10: 1441965947

ISBN-13: 9781441965943

Approximation concept and numerical research are valuable to the production of exact computing device simulations and mathematical types. examine in those parts can impression the computational innovations utilized in quite a few mathematical and computational sciences. This choice of contributed chapters, devoted to the well known mathematician Gradimir V. Milovanović, signify the new paintings of specialists within the fields of approximation thought and numerical research. those invited contributions describe new developments in those vital parts of study together with theoretic advancements, new computational algorithms, and multidisciplinary purposes. distinct positive factors of this quantity: - offers effects and approximation equipment in numerous computational settings together with polynomial and orthogonal platforms, analytic features, and differential equations. - presents a old evaluation of approximation concept and lots of of its subdisciplines. - comprises new effects from diversified components of study spanning arithmetic, engineering, and the computational sciences. "Approximation and Computation" is meant for mathematicians and researchers concentrating on approximation concept and numerical research, yet is additionally a worthwhile source to scholars and researchers in engineering and different computational and utilized sciences.

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0 (z) = 1, 2 αk = ϑk − ϑk−1 , βk = ϑk−1 , k = 1, 2, . . , Γ ((k + 2)/2) 2 2k + 1 Γ ((k + 1)/2) (5) (6) 2 , k ≥ 0. (7) As k → ∞, one finds αk → 0, βk → 14 , familiar from Szeg¨o’s class of polynomials orthogonal on the interval [−1, 1]. Interestingly, the polynomials πn are closely connected to Legendre polynomials, πn (z) = Pˆn (z) − iϑn−1 Pˆn−1(z), n ≥ 1, (8) where Pˆk is the monic Legendre polynomial of degree k. This allowed us to derive a linear second-order differential equation for πn , which, like the differential equation for Legendre polynomials, has regular singular points at 1, −1, and ∞, but unlike Legendre polynomials, an additional singular point on the negative imaginary axis, which depends on n and approaches the origin monotonically as n ↑ ∞.

20 Aleksandar Ivi´c Fig. 1 The constant Cn (α ) for n = 1, 2, 4 and n = ∞ Fig. 2 Enlarged nontrivial part in Fig. 1 GVM [16] constructs Cn (α ) in (9) as ⎧ 1 ⎪ ⎪ ⎪ ⎨ (2 + α )(1 + α ) , Cn (α ) = ⎪ n2 ⎪ ⎪ , ⎩ (2n + α )(2n + α − 1) −1 ≤ α ≤ αn , αn ≤ α < ∞, which is presented in Figs. 1 and 2. 2 Extremal Problems for the Lorentz Class of Polynomials Extremal problems for the Lorentz class of polynomials with respect to the Jacobi weight w(t) = (1 − t)α (1 + t)β , α , β > −1 were investigated by Milovanovi´c and Petkovi´c [32].

Using these inequalities, it is obvious that we have 2(h1 + · · · + hn) < b − a. , b a f (x)w(x) dx = n wk ∑ 2hk k=1 xk +hk xk −hk f (x)w(x) dx + Rn( f ), (3) where Rn ( f ) = 0 for each f ∈ P2n−1 . If hk = h, 1 k n, they also proved the uniqueness of (3). In 2003, they proved the uniqueness of (3) for the Legendre weight (w(x) = 1) for any set of lengths hk 0, k = 1, . . , n, satisfying the condition (2). Recently Milovanovi´c and Cvetkovi´c [104, 107, 110], by using properties of the topological degree of nonlinear mappings, proved that the Gaussian interval quadrature formula is unique for the Jacobi weight function w(x) = (1 − x)α (1 + x)β , α , β > −1, on [−1, 1] and they proposed an algorithm for numerical construction.

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Approximation and Computation: In Honor of Gradimir V. Milovanović by Aleksandar Ivić (auth.), Walter Gautschi, Giuseppe Mastroianni, Themistocles M. Rassias (eds.)

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