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.
into the moment
. Substituting it into (5) gives
Note that the interval mean is simply the midpoint.
Logarithm This weight is present in many physical problems, the main body of which exhibits some axial symmetry.
, the moment
and (5) imply
The optimal point is closer to the origin than the midpoint , reflecting the strength of the log singularity.
, into the moment
Inequality (5) gives .
The optimal point is again shifted to the left of the midpoint due to the singularity at the origin.
Chebyshev Substituting , , , into the moment gives .
Hence, the inequality corresponding to the Chebyshev weight is .
The optimal point is at the midpoint of the interval reflecting the symmetry of the Chebyshev weight over its interval.
Laguerre The Laguerre weight , is defined for positive values, . From the moment , we have .
The appropriate inequality is , from which the optimal sample point of may be deduced.
Hermite Finally, the Hermite weight is defined over the entire real line . The inequality (5) with the Hermite weight function is thus , which results in an optimal sampling point of .