In mathematics, Turán's inequalities are some inequalities for

Legendre polynomials In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...

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

P_n

is the

n

Legendre polynomial In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...

, 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 a ...

, 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 orthogonal 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: ...

they are :

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

References

* * * Orthogonal polynomials Inequalities (mathematics) {{mathanalysis-stub

See also

References