Abstract
It is well known that the interpolation error for left| xright| ^{alpha },alpha >0 in L_{infty }left[ -1,1right] by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind can be represented in its limiting form by entire functions of exponential type. In this paper, we establish new asymptotic bounds for these quantities when alpha tends to infinity. Moreover, we present some explicit constructions for near best approximation polynomials to left| xright| ^{alpha },alpha >0 in the L_{infty } norm which are based on the Chebyshev interpolation process. The resulting formulas possibly indicate a general approach towards the structure of the associated Bernstein constants.
Highlights
It is well known that the interpolation error for |x|α, α > 0 in L∞ [−1, 1] by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind can be represented in its limiting form by entire functions of exponential type
By the homogeneity property, Bernstein established a representation for the quantities Δ∞,α in form of the approximation error on the real line for |x|α by entire functions of exponential type, namely
= inf |x|α − H L∞(R) : H is entire of exponential type ≤ 1 . (1.1)
Summary
Let α > 0 be not an even integer. Starting in year 1913 for the case α = 1, and later in 1938 for the general case α > 0, Bernstein [1,2] established the existence of the limit
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