Approximating Continuous Functions by Polynomials

Using the given data, construct the polynomial $P_{n}(x)$ of lowest degree $n$:

What are $P_{n}(-1)$, $P_{n}(1)$, $P_{n}(6)$ approximately equal to?

Using the Lagrange interpolation formula, write the equation of the parabola

passing through the three points $A(x_{0}-h,y_{-1})$, $B(x_{0},y_{0})$, $C(x_{0}+h,y_{1})$.

Using the values $x_{0}=1$, $y_{0}=1$; $x_{1}=25$, $y_{1}=5$; $x_{2}=100$, $y_{2}=10$, derive an approximate formula for the square root $y=\sqrt{x}$ ($1 \leq x \leq 100$).

Using the values $\sin 0^{\circ }=0$, $\sin 30^{\circ }=\dfrac {1}{2}$, $\sin 90^{\circ }=1$, derive the approximate formula

Using this formula, approximately find: $\sin 20^{\circ }$, $\sin 40^{\circ }$, $\sin 80^{\circ }$.

Taking the values $x_{i}=0,\pm \dfrac {1}{2},\pm 1$ as the interpolation points of a Lagrange polynomial, construct the Lagrange interpolation polynomial on the closed interval $\left[-1,1\right]$ for the function $f(x)=\left|x\right|$.

Using a Lagrange polynomial in place of the function $y(x)$, approximately compute

where

For the functions $x$, $x^{2}$, $x^{3}$, construct on the closed interval $\left[0,1\right]$ the Bernstein polynomial

For a given function $f(x)$ on the closed interval $\left[a,b\right]$, write out the formula for the Bernstein polynomial $B_{n}(x)$.

On the interval $\left[-1,1\right]$, approximate the function $f(x)=\dfrac {\left|x\right|+x}{2}$ using the Bernstein polynomial $B_{4}(x)$. Plot the graphs of $y=\dfrac {\left|x\right|+x}{2}$ and $y=B_{4}(x)$.

On $-1 \leq x \leq 1$, approximate the function using an even-degree Bernstein polynomial

For the function $f(x)=e^{kx}$ ($a \leq x \leq b$), write out the Bernstein polynomial $B_{n}(x)$.

On the closed interval $-\dfrac {\pi }{2} \leq x \leq \dfrac {\pi }{2}$, for the function $f(x)=\cos x$, compute the Bernstein polynomial $B_{n}(x)$.

Prove: on the closed interval $\left[-1,1\right]$, $\left|x\right|=\displaystyle \lim_{n \rightarrow \infty } P_{n}(x)$, where

Let $f(x) \in C\left[a,b\right]$, $M_{k}=\displaystyle \int_{a}^{b} x^{k}f(x)\, dx=0$ ($k=0,1,2,\cdots$). Prove: for $x \in \left[a,b\right]$, $f(x) \equiv 0$.

Let $f(x)$ be a continuous function on $-\pi \leq x \leq \pi$, and let $a_{n}$, $b_{n}$ ($n=0,1,2,\cdots$) be its Fourier coefficients. Prove that the Fejer trigonometric polynomial

converges uniformly to the function $f(x)$ on the interval $(-\pi ,\pi )$.

Construct the Fejer polynomial $\sigma_{2n-1}(x)$ for the following function: