By Bamberg P. G., Sternberg Sh.
After the fundamental theories of differential and fundamental calculus are defined, they're utilized to fascinating difficulties in optics, electronics (networks), electrostatics, wave dynamics and eventually, to classical thermodynamics.
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1; and also the q-binomial coefficient is defined by n k := q [n]q ! [n − k]q ! It is well-known that the polynomials in (1) and (2) are positive and linear, and so their approximation properties can easily be obtain from the classical Korovkin theorem (see ). However, in recent years, a nonlinear modification of the classical Bernstein polynomial has been introduced by Bede and Gal  (see, also, ). Although, of course, the Korovkin theorem fails for this nonlinear operator, they showed in  that the new operator has a similar approximation behavior to the classical Bernstein polynomial.
However, so far, there is no such an improvement on the q-Bernstein polynomial given by (2). The aim of the present paper is to fill in this gap in the literature. This paper is organized as follows. In the second section, we construct a nonlinear q-Bernstein operator of max-product kind, and in the third section, we obtain an error estimation for these operators. In the last section, we give an statistical approximation theorem and discuss some concluding remarks. 2 Construction of the Operators In this section, we construct a nonlinear approximation operator by modifying the q-Bernstein polynomial given by (2).
75) Explanation: for x, y ∈ [−1, 0] we get that 1 − x, 1 − y ≥ 1 , and 0 ≤ h 1 − α1h 1 , h 2 − α2h 2 < 1. Hence (1 − x)h 1 −α1h1 , (1 − y)h 2 −α2h2 ≥ 1, and then (1 − x)h 1 −α1h1 (1 − y)h 2 −α2h2 ≥ 1, so that Hence again 1 − (1 − x)h 1 −α1h1 (1 − y)h 2 −α2h2 ≤ 0. (76) 2 L Q ∗∗ n,m (x, y) ≥ 0, for (x, y) ∈ [−1, 0] . (77) References 1. : Monotone approximation by pseudopolynomials. In: Approximation Theory. Academic Press, New York (1991) 2. : Bivariate monotone approximation. Proc. Amer. Math. Soc.
A Course in Mathematics for Students of Physics: Volume 1 by Bamberg P. G., Sternberg Sh.