In mathematics, Turán's inequalities are some inequalities for

Legendre polynomials In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...

found by (and first published by ). There are many generalizations to other polynomials, often called Turán's inequalities, given by and other authors. If is the th

Legendre polynomial In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...

, Turán's inequalities state that :

\,\! P_n(x)^2 > P_(x)P_(x)\text-1 For ''H''

_''n'', the ''n''th

Hermite polynomial In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as i ...

, Turán's inequalities are :

H_n(x)^2 - H_(x)H_(x)= (n-1)!\cdot \sum_^\fracH_i(x)^2>0 ~,

whilst for

Chebyshev polynomials The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshe ...

they are :

T_n(x)^2 - T_(x)T_(x)= 1-x^2>0 \text-1

References

* * * Orthogonal polynomials Inequalities {{mathanalysis-stub

See also

References