S.N. Bernstein’s Polynomials and the Interpolation Problem

Tiberiu Popoviciu
(proceedings), approximation of univariate functions, Approximation Theory, Bernstein, interpolation, paper

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T. Popoviciu, Les polynomes de S.N. Bernstein et le problème de l’interpolation. Congresul internațional al matematicienilor, Amsterdam, 1954.

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1954 a -Popoviciu- Int. Congress Math. – Les polynomes de S.N. Bernstein et le probleme de l’interpolation

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1954 a -Popoviciu- Int. Congress Math. - SN Bernstein polynomials and the interpo problem
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SN BERNSTEIN'S POLYNOMIALS AND THE PROBLEM OF INTERPOLATION 1954

Tiberiu Popoviciu

  1. The simplest interpolation problems involve substituting for the function f ( x ) f ( x ) f(x)f(x)f(x), defined in the closed interval [ has , b ] [ has , b ] [a,b][a, b][has,b], the function
(1) F ( x ; f ) = ∑ i = 0 n φ i ( x ) f ( x i ) (1) F ( x ; f ) = ∑ i = 0 n   φ i ( x ) f x i {:(1)F(x;f)=sum_(i=0)^(n)varphi_(i)(x)f(x_(i)):}\begin{equation*} F(x ; f)=\sum_{i=0}^{n} \varphi_{i}(x) f\left(x_{i}\right) \tag{1} \end{equation*}(1)F(x;f)=∑i=0nφi(x)f(xi)
where the distinct points (nodes) x i ∈ [ has , b ] x i ∈ [ has , b ] x_(i)in[a,b]x_{i} \in[a, b]xi∈[has,b]and functions φ i ( x ) φ i ( x ) varphi_(i)(x)\varphi_{i}(x)φi(x), defined in [ has , b ] [ has , b ] [a,b][a, b][has,b]are independent of the function f ( x ) f ( x ) f(x)f(x)f(x).
We can pose the problem of determining and studying the processes (1), which not only remain non-negative for all f ( x ) f ( x ) f(x)f(x)f(x), non-negative, but which also retain certain properties relating to the shape of the function f ( x ) f ( x ) f(x)f(x)f(x). E.g., which preserve the monotonicity, convexity, etc., of the function f ( x ) f ( x ) f(x)f(x)f(x).
2. If the function (1), not only remains non-negative for all f ( x ) f ( x ) f(x)f(x)f(x)non-negative, throughout the range [ has , b ] [ has , b ] [a,b][a, b][has,b], but keeps in [ has , b ] [ has , b ] [a,b][a, b][has,b]also any convexity property of any order of the function f ( x ) f ( x ) f(x)f(x)f(x), THE φ i ( x ) φ i ( x ) varphi_(i)(x)\varphi_{i}(x)φi(x)are polynomials of the degree ≦ n ≦ n <= n\leqq n≦nand, more precisely, F ( x ; f ) F ( x ; f ) F(x;f)F(x ; f)F(x;f)is a polynomial of the degree ≦ k ≦ k <= k\leqq k≦kfor any polynomial f ( x ) f ( x ) f(x)f(x)f(x)of the degree k k kkk, For k = 0 , 1 , … k = 0 , 1 , … k=0,1,dotsk=0.1, \ldotsk=0,1,…. Furthermore, it is necessary and sufficient that the inequalities
∑ i = 0 n ( | x i − λ | + x i − λ ) s φ i ( s + 1 ) ( x ) ≧ 0 , For x ∈ [ has , b ] , s = 0 , 1 , … ∑ i = 0 n   x i − λ + x i − λ s φ i ( s + 1 ) ( x ) ≧ 0 ,  For  x ∈ [ has , b ] , s = 0 , 1 , … sum_(i=0)^(n)(|x_(i)-lambda|+x_(i)-lambda)^(s)varphi_(i)^((s+1))(x) >= 0," for "x in[a,b],s=0,1,dots\sum_{i=0}^{n}\left(\left|x_{i}-\lambda\right|+x_{i}-\lambda\right)^{s} \varphi_{i}^{(s+1)}(x) \geqq 0, \text { for } x \in[a, b], s=0,1, \ldots∑i=0n(|xi−λ|+xi−λ)sφi(s+1)(x)≧0, For x∈[has,b],s=0,1,…
are verified [3].
Let us also note that if { F n ( x ; f ) } F n ( x ; f ) {F_(n)(x;f)}\left\{F_{n}(x ; f)\right\}{Fn(x;f)}is an infinite sequence of functions of the form (1), which enjoy the property of conservation of the shape specified above, and if this sequence tends in [ has , b ] [ has , b ] [a,b][a, b][has,b]towards f ( x ) f ( x ) f(x)f(x)f(x), For f ( x ) = 1 , x , x 2 f ( x ) = 1 , x , x 2 f(x)=1,x,x^(2)f(x)=1, x, x^{2}f(x)=1,x,x2, then the sequence converges absolutely and uniformly to f ( x ) f ( x ) f(x)f(x)f(x)In [ has , b ] [ has , b ] [a,b][a, b][has,b], when f ( x ) f ( x ) f(x)f(x)f(x)is continuous in this interval.
3. The simplest process of the form (I) is constituted by the SN Bernstein polynomials [1]
(2) B n ( x ; f ) = ∑ i = 0 n ( n i ) + ( i n ) x i ( 1 − x ) n − i ( [ has , b ] = [ 0 , 1 ] ) (2) B n ( x ; f ) = ∑ i = 0 n   ( n i ) + i n x i ( 1 − x ) n − i ( [ has , b ] = [ 0 , 1 ] ) {:(2)B_(n)(x;f)=sum_(i=0)^(n)((n)/(i))+((i)/(n))x^(i)(1-x)^(ni)quad([a","b]=[0","1]):}\begin{equation*} B_{n}(x; f)=\sum_{i=0}^{n}\binom{n}{i}+\left(\frac{i}{n}\right) x^{i}(1-x)^{ni} \quad([a, b]=[0,1]) \tag{2} \end{equation*}(2)Bn(x;f)=∑i=0n(ni)+(in)xi(1−x)n−i([has,b]=[0,1])
I have previously demonstrated [2] that these polynomials retain all convexity properties of the function f ( x ) f ( x ) f(x)f(x)f(x). But there are more complete properties. The formula which gives the derivative of the polynomial (2),
B n ′ ( x ; f ) = n ∑ i = 0 n − 1 Δ 1 n f ( i n ) ( n − 1 i ) x i ( 1 − x ) n − 1 − i B n ′ ( x ; f ) = n ∑ i = 0 n − 1   Δ 1 n f i n ( n − 1 i ) x i ( 1 − x ) n − 1 − i B_(n)^(')(x;f)=nsum_(i=0)^(n-1)Delta_((1)/(n))f((i)/(n))((n-1)/(i))x^(i)(1-x)^(n-1-i)B_{n}^{\prime}(x ; f)=n \sum_{i=0}^{n-1} \Delta_{\frac{1}{n}} f\left(\frac{i}{n}\right)\binom{n-1}{i} x^{i}(1-x)^{n-1-i}Bn′(x;f)=n∑i=0n−1Δ1nf(in)(n−1i)xi(1−x)n−1−i
shows us that if f ( x ) f ( x ) f(x)f(x)f(x)is formed by m m mmmmonotone pieces (is segmentally monotone), the polynomial (2) enjoys the same property. In general, s s sssbeing a natural number or 0, the derivative of order s + 1 s + 1 s+1s+1s+1,
B n ( s + 1 ) ( x ; f ) = n ( n − 1 ) … ( n − s ) ∑ i = 0 n − s − 1 Δ n 1 s + 1 f ( i n ) ( n − s − 1 i ) x i ( 1 − x ) n − s − 1 − i B n ( s + 1 ) ( x ; f ) = n ( n − 1 ) … ( n − s ) ∑ i = 0 n − s − 1   Δ n 1 s + 1 f i n ( n − s − 1 i ) x i ( 1 − x ) n − s − 1 − i B_(n)^((s+1))(x;f)=n(n-1)dots(ns)sum_(i=0)^(ns-1)Delta_((n)/(1))^(s+1)f((i)/(n))((ns-1)/(i))x^(i)(1-x)^(ns-1-i)B_{n}^{(s+1)}(x ; f)=n(n-1) \ldots(ns) \sum_{i=0}^{ns-1} \Delta_{\frac{n}{1}}^{s+1} f\left(\frac{i}{n}\right)\binom{ns-1}{i} x^{i}(1-x)^{ns-1-i}Bn(s+1)(x;f)=n(n−1)…(n−s)∑i=0n−s−1Δn1s+1f(in)(n−s−1i)xi(1−x)n−s−1−i
shows us that if f ( x ) f ( x ) f(x)f(x)f(x)is formed by m m mmmpieces of non-concave or non-convex functions of order s s sss, the polynomial (2) is formed by at most ( m − 1 ) ( s + 2 ) ( m − 1 ) ( s + 2 ) (m-1)(s+2)(m-1)(s+2)(m−1)(s+2)+1 similar pieces.
Using the properties of order functions n n nnnby segments [3] we can still complete these properties.
4. Interpolation processes which preserve the properties of convexity and, in general, certain shapes of the function f ( x ) f ( x ) f(x)f(x)f(x), also have a practical interest. It is important that in the graphical representation of a function arising from a practical problem, the discrete points, obtained by observations, can be used to approximately represent the variation of the phenomenon studied by a curve having a theoretically predicted appearance.

Bibliography

[1] SN Bernstein, Comm. Charkow, (2):13, 1-2 (1912) or Sobranie Socinenie, I, 105.
[2] T. Popoviciu, Mathematica, 10, 45-53 (1934). 419-54
[3] T. Popoviciu, Bull. Acad. Romanian, XXIV, 409-416 (1942).
Filiala Academiei RPR, Str. Pavlov 27 Cluj., RPR (Romania).
1954

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