Integration with weight functions is used in countless mathematical problems. Two main areas are: (i) approximation theory and spectral analysis and (ii) statistical analysis and the theory of distributions. In this section, inequality (5) is evaluated for the more popular weight functions.

**Uniform (Legender)** Substituting

$w(t)=1$ into the moment

$\sigma (a,b)=\frac{M(a,b)}{m(a,b)}$ gives

$\sigma (a,b)=\frac{a+b}{2}$. Substituting it into (5) gives

Note that the interval mean $\sigma (a,b)$ is simply the midpoint.

**Logarithm** This weight is present in many physical problems, the main body of which exhibits some axial symmetry.

Putting

$w(t)=ln\frac{1}{t}$,

$a=0$,

$b=1$, the moment

$\sigma (a,b)=\frac{M(a,b)}{m(a,b)}$ and (5) imply

The optimal point $\sigma (0,1)=\frac{1}{4}$ is closer to the origin than the midpoint $\sigma (a,b)=\frac{a+b}{2}$, reflecting the strength of the log singularity.

**Jacobi** Substituting

$w(t)=\frac{1}{\sqrt{t}}$,

$a=0$,

$b=1$, into the moment

$\sigma (a,b)=\frac{M(a,b)}{m(a,b)}$ gives

$\sigma (0,1)=\frac{{\int}_{0}^{1}t\frac{1}{\sqrt{t}}\phantom{\rule{0.2em}{0ex}}dt}{{\int}_{0}^{1}\frac{1}{\sqrt{t}}\phantom{\rule{0.2em}{0ex}}dt}=\frac{1}{3}.$

(18)

Inequality (5) gives $|f(x)-\frac{1}{2}{\int}_{0}^{1}f(t)\frac{1}{\sqrt{t}}\phantom{\rule{0.2em}{0ex}}dt-(x-\frac{1}{3}){f}^{\mathrm{\prime}}(x)|\le \frac{1}{4}(\varphi -\theta )(1+\frac{1}{2}|{\int}_{0}^{1}sgn(t-x)\frac{1}{\sqrt{t}}\phantom{\rule{0.2em}{0ex}}dt|)$.

The optimal point $\sigma (0,1)=\frac{1}{3}$ is again shifted to the left of the midpoint due to the $\frac{1}{\sqrt{t}}$ singularity at the origin.

**Chebyshev** Substituting $w(t)=\frac{1}{\sqrt{1-{t}^{2}}}$, $a=-1$, $b=1$, into the moment $\sigma (a,b)=\frac{M(a,b)}{m(a,b)}$ gives $\sigma (-1,1)=\frac{{\int}_{-1}^{1}t\frac{1}{\sqrt{1-{t}^{2}}}\phantom{\rule{0.2em}{0ex}}dt}{{\int}_{-1}^{1}\frac{1}{\sqrt{1-{t}^{2}}}\phantom{\rule{0.2em}{0ex}}dt}=0$.

Hence, the inequality corresponding to the Chebyshev weight is $|f(x)-\frac{1}{m(a,b)}{\int}_{-1}^{1}f(t)\times \frac{1}{\sqrt{1-{t}^{2}}}\phantom{\rule{0.2em}{0ex}}dt-x{f}^{\mathrm{\prime}}(x)|\le \frac{1}{4}(\varphi -\theta )(\frac{\pi}{2}+\frac{1}{2}|{\int}_{-1}^{1}sgn(t-x)\frac{1}{\sqrt{1-{t}^{2}}}\phantom{\rule{0.2em}{0ex}}dt|)$.

The optimal point is at the midpoint of the interval reflecting the symmetry of the Chebyshev weight over its interval.

**Laguerre** The Laguerre weight $w(t)={e}^{-t}$, is defined for positive values, $t\in [0,\mathrm{\infty})$. From the moment $\sigma (a,b)=\frac{M(a,b)}{m(a,b)}$, we have $\sigma (0,\mathrm{\infty})=\frac{{\int}_{0}^{\mathrm{\infty}}t{e}^{-t}\phantom{\rule{0.2em}{0ex}}dt}{{\int}_{0}^{\mathrm{\infty}}{e}^{-t}\phantom{\rule{0.2em}{0ex}}dt}=1$.

The appropriate inequality is $|f(x)-{\int}_{0}^{\mathrm{\infty}}f(t){e}^{-t}\phantom{\rule{0.2em}{0ex}}dt-(x-1){f}^{\mathrm{\prime}}(x)|\le \frac{1}{4}(\varphi -\theta )(\frac{1}{2}+\frac{1}{2}|{\int}_{0}^{\mathrm{\infty}}sgn(t-x){e}^{-t}\phantom{\rule{0.2em}{0ex}}dt|)$, from which the optimal sample point of $x=1$ may be deduced.

**Hermite** Finally, the Hermite weight is $w(t)={e}^{-{t}^{2}}$ defined over the entire real line $\sigma (-\mathrm{\infty},\mathrm{\infty})=\frac{{\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}t{e}^{-{t}^{2}}\phantom{\rule{0.2em}{0ex}}dt}{{\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}{e}^{-{t}^{2}}\phantom{\rule{0.2em}{0ex}}dt}=0$. The inequality (5) with the Hermite weight function is thus $|f(x)-\frac{1}{\pi}{\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}f(t){e}^{-{t}^{2}}\phantom{\rule{0.2em}{0ex}}dt-x{f}^{\mathrm{\prime}}(x)|\le \frac{1}{4}(\varphi -\theta )(\frac{\pi}{2}+\frac{1}{2}|{\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}sgn(t-x){e}^{-{t}^{2}}\phantom{\rule{0.2em}{0ex}}dt|)$, which results in an optimal sampling point of $x=0$.