On the approximations of continuous functions of a real variable by polynomials

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T. Popoviciu, Sur les l’approximations des fonctions continues d’une variable réelle par des polynomes, Ann. Sci. Univ. Iassy, 28 (1942) p. 208 (in French).

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Ann. Sci. Univ. Iassy

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[MR0018270]

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ON THE APPROXIMATION OF CONTINUOUS FUNCTIONS OF A REAL VARIABLE BY POLYNOMIES

by

TIBERIU POPOVICIU

In a previous work 1 1 ^(1){ }^{1}1) we have shown that the MS Bernstein polynomials
P n ( x ; f ) = ∑ i = 0 n f ( i n ) ( n i ) x i ( 1 − x ) n − i P n ( x ; f ) = ∑ i = 0 n   f i n ( n i ) x i ( 1 − x ) n − i P_(n)(x;f)=sum_(i=0)^(n)f((i)/(n))((n)/(i))x^(i)(1-x)^(ni)P_{n}(x ; f)=\sum_{i=0}^{n} f\left(\frac{i}{n}\right)\binom{n}{i} x^{i}(1-x)^{ni}Pn(x;f)=∑i=0nf(in)(ni)xi(1−x)n−i
give, for the function f ( x ) f ( x ) f(x)f(x)f(x)continues in the closed interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1], an approximation of the order of ω ( 1 n ) ω 1 n omega((1)/(sqrtn))\omega\left(\frac{1}{\sqrt{n}}\right)ω(1n)if ω ( δ ) ω ( δ ) omega (delta)omega(delta)ω(δ)is the oscillation modulus of f ( x ) f ( x ) f(x)f(x)f(x)Here is a simpler demonstration of this result.
If we notice that ∑ i = 0 n ( n i ) x i ( 1 − x ) n − i = 1 ∑ i = 0 n   ( n i ) x i ( 1 − x ) n − i = 1 sum_(i=0)^(n)((n)/(i))x^(i)(1-x)^(ni)=1\sum_{i=0}^{n}\binom{n}{i} x^{i}(1-x)^{ni}=1∑i=0n(ni)xi(1−x)n−i=1the known properties of ω ( δ ) ω ( δ ) omega (delta)omega(delta)ω(δ)give us
| f ( x ) − P n ( x ; f ) | ≤ ( 1 δ ∑ i = 0 n | x − i n | ( n i ) x i ( 1 − x ) n − i + 1 ) ω { δ } , x ∈ [ 0 , 1 ] . f ( x ) − P n ( x ; f ) ≤ 1 δ ∑ i = 0 n &hairsp; x − i n ( n i ) x i ( 1 − x ) n − i + 1 ω { δ } , x ∈ [ 0 , 1 ] . |f(x)-P_(n)(x;f)| <= ((1)/(delta)sum_(i=0)^(n)|x-(i)/(n)|((n)/(i))x^(i)(1-x)^(ni)+1)omega{delta},quad x in[0,1].\left|f(x)-P_{n}(x ; f)\right| \leq\left(\frac{1}{\delta} \sum_{i=0}^{n}\left|x-\frac{i}{n}\right|\binom{n}{i} x^{i}(1-x)^{ni}+1\right) \omega\{\delta\}, \quad x \in[0,1] .|f(x)−Pn(x;f)|≤(1δ∑i=0n|x−in|(ni)xi(1−x)n−i+1)ω{δ},x∈[0,1].
But we have, by applying a well-known inequality,
∑ i = 0 n | x − i n | ( n i ) x i ( 1 − x ) n − i ≤ ∑ i = 0 n ( x − i n ) 2 ( n i ) x i ( 1 − x ) n − i = = x ( 1 − x ) n ≤ 1 2 n , x ∈ [ 0 , 1 ] ∑ i = 0 n &hairsp; x − i n ( n i ) x i ( 1 − x ) n − i ≤ ∑ i = 0 n &hairsp; x − i n 2 ( n i ) x i ( 1 − x ) n − i = = x ( 1 − x ) n ≤ 1 2 n , x ∈ [ 0 , 1 ] {:[sum_(i=0)^(n)|x-(i)/(n)|((n)/(i))x^(i)(1-x)^(ni) <= sqrt(sum_(i=0)^(n)(x-(i)/(n))^(2)((n)/(i))^(x^(i)(1-x)^(ni)))=],[=sqrt((x(1-x))/(n)) <= (1)/(2sqrtn)","quad x in[0","1]]:}\begin{gathered} \sum_{i=0}^{n}\left|x-\frac{i}{n}\right|\binom{n}{i} x^{i}(1-x)^{ni} \leq \sqrt{\sum_{i=0}^{n}\left(x-\frac{i}{n}\right)^{2}\binom{n}{i}^{x^{i}(1-x)^{ni}}}= \\ =\sqrt{\frac{x(1-x)}{n}} \leq \frac{1}{2 \sqrt{n}}, \quad x \in[0,1] \end{gathered}∑i=0n|x−in|(ni)xi(1−x)n−i≤∑i=0n(x−in)2(ni)xi(1−x)n−i==x(1−x)n≤12n,x∈[0,1]
By taking δ = 1 n δ = 1 n delta = (1)/(sqrtn)\delta=\frac{1}{\sqrt{n}}δ=1n, We have
| f ( x ) − P n ( x ; f ) | ≤ 3 2 ω ( 1 n ) f ( x ) − P n ( x ; f ) ≤ 3 2 ω 1 n |f(x)-P_(n)(x;f)| <= (3)/(2)omega((1)/(sqrtn))\left|f(x)-P_{n}(x ; f)\right| \leq \frac{3}{2} \omega\left(\frac{1}{\sqrt{n}}\right)|f(x)−Pn(x;f)|≤32ω(1n)
which demonstrates ownership.

    1. Tiberiu Popoviciu. "On the approximation of higher-order convex functions" Mathematica, 10, 49-54, (1934).
1942

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