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On approximation of functions by certain operators preserving x 2

Lucyna Rempulska, Karolina Tomczak (2008)

Commentationes Mathematicae Universitatis Carolinae

In this paper we extend the Duman-King idea of approximation of functions by positive linear operators preserving e k ( x ) = x k , k = 0 , 2 . Using a modification of certain operators L n preserving e 0 and e 1 , we introduce operators L n * which preserve e 0 and e 2 and next we define operators L n ; r * for r -times differentiable functions. We show that L n * and L n ; r * have better approximation properties than L n and L n ; r .

On Bernstein inequalities for multivariate trigonometric polynomials in L p , 0 ≤ p ≤ ∞

Laiyi Zhu, Xingjun Zhao (2022)

Czechoslovak Mathematical Journal

Let 𝕋 n be the space of all trigonometric polynomials of degree not greater than n with complex coefficients. Arestov extended the result of Bernstein and others and proved that ∥ ( 1 / n ) T n ' ∥ p ≤ ∥ T n ∥ p for 0 ≤ p ≤ ∞ and T n ∈ 𝕋 n . We derive the multivariate version of the result of Golitschek and Lorentz T n cos α + 1 n ∇ T n sin α l ∞ ( m ) p ≤ ∥ T n ∥ p , 0 ≤ p ≤ ∞ for all trigonometric polynomials (with complex coeffcients) in m variables of degree at most n .

On best approximation in fuzzy metric spaces

Naser Abbasi, Hamid Mottaghi Golshan (2015)

Kybernetika

In this paper we introduce the notation of t-best approximatively compact sets, t-best approximation points, t-proximinal sets, t-boundedly compact sets and t-best proximity pair in fuzzy metric spaces. The results derived in this paper are more general than the corresponding results of metric spaces, fuzzy metric spaces, fuzzy normed spaces and probabilistic metric spaces.

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