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T. Popoviciu, Asupra demonstraţiei teoremei lui Weierstrass cu ajutorul polinoamelor de interpolare, Lucrările ses. generale a Acad. R.P.R., pp. 1664-1667 (in Romanian).
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republished in English, in 1998: T. Popoviciu, On the proof of Weierstrass’ theorem using interpolation polynomials, East J. Approximations, 4 (1998) no. 1, pp. 107-110 (translated by D. Kacsó)
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ON THE PROOF OF WEIERSTRASS'S THEOREM WITH THE HELP OF INTERPOLATION POLYNOMIALS
I. Either[a,b][a, b]an interval ends and closed, which, as the case may be, can be considered reduced to[0,I][0, \mathrm{I}]or[-I,I][-\mathrm{I}, \mathrm{I}], which does not restrict the generality of the issues addressed.
Let us consider a triangular array of nodes in[a,b][a, b]
{:(I)x_(n)","o","x_(n,1)","dots","x_(n,n)","quad n=1","2","3","dots:}\begin{equation*} x_{n}, o, x_{n, 1}, \ldots, x_{n, n}, \quad n=1,2,3, \ldots \tag{I} \end{equation*}
and a triangular array of polynomials inxx,
{:(2)P_(n,0)","quadP_(n,1)","dots","P_(n,n)","quad n=1","2","3","dots(P_(n,m)=P_(n,m)(x)):}\begin{equation*} P_{n, 0}, \quad P_{n, 1}, \ldots, P_{n, n}, \quad n=1,2,3, \ldots\left(P_{n, m}=P_{n, m}(x)\right) \tag{2} \end{equation*}
A functionf(x)f(x)uniform[a,b][a, b]attach the side of polynomial of interpolation (in a general sense)
Q_(n)[f;x]=sum_(i=0)^(n)P_(n,t)(x)f(x_(n,t)),quad n=1,2,3,dotsQ_{n}[f ; x]=\sum_{i=0}^{n} P_{n, t}(x) f\left(x_{n, t}\right), \quad n=1,2,3, \ldots
The problem of determining the arrays (1) and (2) such that
lim_(n rarr oo)Q_(n)[f;x]=f(x)\lim _{n \rightarrow \infty} Q_{n}[f ; x]=f(x)
uniform in[a,b][a, b]for any functionf(x)f(x)continue in[a,b][a, b], put byAND\mathbb{E}. Bore1 [I], a most elegant solution of S. Bernstein [2] by polynomials (written for the interval[0,I][0, I]).
B_(n)[f;x]=sum_(i=0)^(n)f((i)/(n))((n)/(i))x^(i)(1-x)^(nl)B_{n}[f ; x]=\sum_{i=0}^{n} f\left(\frac{i}{n}\right)\left(\frac{n}{i}\right) x^{i}(1-x)^{nl}
- If in particular we have
a)P_(n,t)(x) >= 0,i=0,1,dots,n;n=1,2,3,dots;x in[a,b]P_{n, t}(x) \geqq 0, i=0.1, \ldots, n ; n=1,2,3, \ldots ; x \in[a, b]
- sum_(i=0)^(n_(i))P_(n,i)(x)=1,quad n=1,2,3,dots\sum_{i=0}^{n_{i}} P_{n, i}(x)=1, \quad n=1,2,3, \ldots
differenceQ_(n)[f;x]-f(x)Q_{n}[f ; x]-f(x)is immediately delimited by the formula
|Q_(n)[f;x]-f(x)| <= 2omega(A_(n))\left|Q_{n}[f ; x]-f(x)\right| \leqq 2 \omega\left(A_{n}\right)
under
A_(n)=su p_([a,b])sum_(i=0)^(r)|x_(n,i)-x|P_(n,i)(x)A_{n}=\sup _{[a, b]} \sum_{i=0}^{r}\left|x_{n, i}-x\right| P_{n, i}(x)
ifomega(delta)\omega(\delta)This oscillating module and functionf(x)f(x)If
​A_(n)rarr0A_{n} \rightarrow 0, forn rarr on \rightarrow \infty, the problem of him E. Bo re 1 is solved. This occurs in particular in the case of S. Bernstein polynomials, as I have shown elsewhere [3].
​A_(n)rarr0A_{n} \rightarrow 0, forn rarr on \rightarrow \infty, the problem of him E. Bo re 1 is solved. This occurs in particular in the case of S. Bernstein polynomials, as I have shown elsewhere [3].
To prove the Weierstrass theorem, in addition to the conditionsalpha\alpha),beta\beta), it is enough to have
(A_(n) <= )B_(n)=sqrt(su p_([a,b])sum_(i=0)^(n)(x_(n,i)-x)^(2)P_(n,i)(x))rarr0," pentra "n rarr oo\left(A_{n} \leqq\right) B_{n}=\sqrt{\sup _{[a, b]} \sum_{i=0}^{n}\left(x_{n, i}-x\right)^{2} P_{n, i}(x)} \rightarrow 0, \text { pentru } n \rightarrow \infty
This is what happened and it was made by S. Bernstein, this is what happenedB_(n)=(I)/(2sqrtn)B_{n}=\frac{I}{2 \sqrt{n}}
3. Weierstrass's theorem can also be proven using the polynomials introduced byI_(1)I_{1}. Headrr[4] and which derives from the interpolation polynomials of him Hermit.
3. Weierstrass's theorem can also be proven using the polynomials introduced byI_(1)I_{1}. Headrr[4] and which derives from the interpolation polynomials of him Hermit.
If
l_(n)(x)=c(x-x_(n,0))(x-x_(n,t))dots(x-x_(n,n)),l_{n}(x)=c\left(x-x_{n, 0}\right)\left(x-x_{n, t}\right) \ldots\left(x-x_{n, n}\right),
underccThis is constant and suitable.
Let us consider the Hermit polynomial
Let us consider the Hermit polynomial
{:(3)sum_(i=0)^(n)h_(n,t)(x)f_(n,i)+sum_(i=0)^(n)k_(n,t)(x)f_(n,t)^('):}\begin{equation*}
\sum_{i=0}^{n} h_{n, t}(x) f_{n, i}+\sum_{i=0}^{n} k_{n, t}(x) f_{n, t}^{\prime} \tag{3}
\end{equation*}
of degree2n+12 n+1, which reduces tof_(n,t)f_{n, t}forx=x_(n,t)x=x_{n, t}and whose derivative: it reduces tof_(n,i)^(')f_{n, i}^{\prime}forx=x_(n,i),i=0,1,dots,nx=x_{n, i}, i=0,1, \ldots, n.
Then the conditionbeta\beta) is achieved throughsum_(i=0)^(n)h_(n,i)(x)=1\sum_{i=0}^{n} h_{n, i}(x)=1
We then have
sum_(i=0)^(n)h_(n,t)(x)(x-x_(n,t))^(2)=2sum_(i=0)^(n)k_(n,t)(x)(x-x_(n,t))=2l_(n)^(2)(x)sum_(i=0)^(n)(1)/([l_(n)^(')(x_(n),t)]^(2))\sum_{i=0}^{n} h_{n, t}(x)\left(x-x_{n, t}\right)^{2}=2 \sum_{i=0}^{n} k_{n, t}(x)\left(x-x_{n, t}\right)=2 l_{n}^{2}(x) \sum_{i=0}^{n} \frac{1}{\left[l_{n}^{\prime}\left(x_{n}, t\right)\right]^{2}}
providedalpha\alpha) is achieved if we takeh_(n,i)(x)=P_(n,i)(x)h_{n, i}(x)=P_{n, i}(x)as shown. L. Fejér, in the particular case for the Ceb âşev polynomial (written for the interval[-I,I][-\mathbf{I}, \mathbf{I}]),
l_(n)(x)=T_(n+1)(x)=cos(n+1)arc cos xl_{n}(x)=T_{n+1}(x)=\cos (n+1) \operatorname{arc} \cos x
In this case
sum_(i=0)^(n)(I)/([l_(n)^(')(x_(n),i)]^(2))=sum_(i=0)^(n)(I)/([T_(n+1)^(')(x_(n,i))]^(2))=(I)/((n+I)^(2))sum_(i=0)^(n)(I-x_(n,i)^(2))=(I)/(2(n+I))\sum_{i=0}^{n} \frac{\mathrm{I}}{\left[l_{n}^{\prime}\left(x_{n}, i\right)\right]^{2}}=\sum_{i=0}^{n} \frac{\mathrm{I}}{\left[T_{n+1}^{\prime}\left(x_{n, i}\right)\right]^{2}}=\frac{\mathrm{I}}{(n+\mathrm{I})^{2}} \sum_{i=0}^{n}\left(\mathrm{I}-x_{n, i}^{2}\right)=\frac{\mathrm{I}}{2(n+\mathrm{I})}
It follows that in this case
B_(n)=(1)/(sqrt(n+1))B_{n}=\frac{1}{\sqrt{n+1}}
So, for L. Fejér's polynomials,
F_(n)[t;x]=sum_(i=0)^(n)h_(n,i)(x)f(x_(n,i))F_{n}[t ; x]=\sum_{i=0}^{n} h_{n, i}(x) f\left(x_{n, i}\right)
underh_(n,t)(x)h_{n, t}(x)are the fundamental polynomials of the first kind corresponding to Hermite polynomials (3) and Chebâşev knotsx_(n,i)x_{n, i}, check the inequality
|F_(n)[f;x]-f(x)| <= 2omega((I)/(sqrt(n+I))),quad x in[-I,I]\left|F_{n}[f ; x]-f(x)\right| \leqq 2 \omega\left(\frac{\mathrm{I}}{\sqrt{n+\mathrm{I}}}\right), \quad x \in[-\mathrm{I}, \mathrm{I}]
It follows from here that the approximation given by L. F ej é r's polynomials is at least of the same order as the approximation given by S. Bexnstein's polynomials.
The above observations raise various new problems on interpolation polynomials, which we will treat in a more detailed paper.
SUMMARY
Delay non-slip locks for approximation polynomialssum_(i=0)^(n)P_(n,i)(x)f(x_(n),i)\sum_{i=0}^{n} P_{n, i}(x) f\left(x_{n}, i\right), generalizing the polynomials of S. N. Bernstein (2). Along with the conditionsalpha\alpha),beta\beta) very light organization used by the cable module functionsf(x)f(x). It is shown that L. Fejer's polynomials (4) lead to Weierstrass's theorem with an approximation reaching the approximation of S. N. Bernstein's polynomials.
SUMMARY
Some remarks on approximation polynomials of the formsum_(sum)^(n)P_(n,i)(x)f(x_(n),i)\sum_{\sum}^{n} P_{n, i}(x) f\left(x_{n}, i\right), generalizing the polynomials of S. Bernstein [2]. Under the conditionsalpha\alpha),beta\beta) we can easily delimit the error using the oscillation modulus of the functionf(x)f(x). As an application it is shown that the polynomials of L. Fejér [4] lead to the Weierstrass theorem, with an approximation which reaches that of the polynomials of S. Bernstein.
BIBLIOGRAPHY
- E. Borel, Lessons on functions of real variables. 1905.
- S. Bernstein, Proof of Weierstrass's theorem based on the calculus of probabilities. Communication Soc. Matem. of Kharkov, 2, 13, 1-2, 1912.
- T. Popovieiu, On the approximation of convex functions of higher order. Mathematica, 10, 49-54, 1934.
- L. Fejér, On Weierstrass approximation, especially by Hermitian interpolation. Math. Ann., 102, 707-725, 1930.