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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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SN BERNSTEIN'S POLYNOMIALS AND THE PROBLEM OF INTERPOLATION 1954
- The simplest interpolation problems involve substituting for the functionf(x)f(x), defined in the closed interval[a,b][a, b], the function
{:(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*}
where the distinct points (nodes)x_(i)in[a,b]x_{i} \in[a, b]and functionsvarphi_(i)(x)\varphi_{i}(x), defined in[a,b][a, b]are independent of the functionf(x)f(x).
We can pose the problem of determining and studying the processes (1), which not only remain non-negative for allf(x)f(x), non-negative, but which also retain certain properties relating to the shape of the functionf(x)f(x). E.g., which preserve the monotonicity, convexity, etc., of the functionf(x)f(x).
2. If the function (1), not only remains non-negative for allf(x)f(x)non-negative, throughout the range[a,b][a, b], but keeps in[a,b][a, b]also any convexity property of any order of the functionf(x)f(x), THEvarphi_(i)(x)\varphi_{i}(x)are polynomials of the degree<= n\leqq nand, more precisely,F(x;f)F(x ; f)is a polynomial of the degree<= k\leqq kfor any polynomialf(x)f(x)of the degreekk, Fork=0,1,dotsk=0.1, \ldots. Furthermore, it is necessary and sufficient that the inequalities
2. If the function (1), not only remains non-negative for allf(x)f(x)non-negative, throughout the range[a,b][a, b], but keeps in[a,b][a, b]also any convexity property of any order of the functionf(x)f(x), THEvarphi_(i)(x)\varphi_{i}(x)are polynomials of the degree<= n\leqq nand, more precisely,F(x;f)F(x ; f)is a polynomial of the degree<= k\leqq kfor any polynomialf(x)f(x)of the degreekk, Fork=0,1,dotsk=0.1, \ldots. Furthermore, it is necessary and sufficient that the inequalities
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
are verified [3].
Let us also note that if{F_(n)(x;f)}\left\{F_{n}(x ; f)\right\}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[a,b][a, b]towardsf(x)f(x), Forf(x)=1,x,x^(2)f(x)=1, x, x^{2}, then the sequence converges absolutely and uniformly tof(x)f(x)In[a,b][a, b], whenf(x)f(x)is continuous in this interval.
3. The simplest process of the form (I) is constituted by the SN Bernstein polynomials [1]
Let us also note that if{F_(n)(x;f)}\left\{F_{n}(x ; f)\right\}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[a,b][a, b]towardsf(x)f(x), Forf(x)=1,x,x^(2)f(x)=1, x, x^{2}, then the sequence converges absolutely and uniformly tof(x)f(x)In[a,b][a, b], whenf(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)=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*}
I have previously demonstrated [2] that these polynomials retain all convexity properties of the functionf(x)f(x). But there are more complete properties. The formula which gives the derivative of the polynomial (2),
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}
shows us that iff(x)f(x)is formed bymmmonotone pieces (is segmentally monotone), the polynomial (2) enjoys the same property. In general,ssbeing a natural number or 0, the derivative of orders+1s+1,
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}
shows us that iff(x)f(x)is formed bymmpieces of non-concave or non-convex functions of orderss, the polynomial (2) is formed by at most(m-1)(s+2)(m-1)(s+2)+1 similar pieces.
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}
shows us that iff(x)f(x)is formed bymmpieces of non-concave or non-convex functions of orderss, the polynomial (2) is formed by at most(m-1)(s+2)(m-1)(s+2)+1 similar pieces.
Using the properties of order functionsnnby segments [3] we can still complete these properties.
4. Interpolation processes which preserve the properties of convexity and, in general, certain shapes of the functionf(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.
4. Interpolation processes which preserve the properties of convexity and, in general, certain shapes of the functionf(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).
[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).