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We study the classical Hermite–Hadamard inequality in the matrix setting. This leads to a number of interesting matrix inequalities such as the Schatten p-norm estimates

$$ \begin{align*}\left(\|A^q\|_p^p + \|B^q\|_p^p\right)^{1/p} \le \|(xA+(1-x)B)^q\|_p+ \|((1-x)A+xB)^q\|_p, \end{align*} $$

for all positive (semidefinite)

$n\times n$ matrices

$A,B$ and

$0<q,x<1$ . A related decomposition, with the assumption

$X^*X+Y^*Y=XX^*+YY^*=I$ , is

$$ \begin{align*}(X^*AX+Y^*BY)\oplus (Y^*AY+X^*BX) =\frac{1}{2n}\sum_{k=1}^{2n} U_k (A\oplus B)U_k^*, \end{align*} $$

for some family of

$2n\times 2n$ unitary matrices

$U_k$ . This is a majorization which is obtained by using the Hansen–Pedersen trace inequality.

In this paper, we prove that any sequence of polynomials (pn)n for which dgr(pn) = n which satisfies a (2N + l)-term recurrence relation is orthogonal with respect to a positive definite N × N matrix of measures. We use that result to prove asymptotic properties of the kernel polynomials associated to a positive measure or a positive definite matrix of measures. Finally, some examples are given.

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Matrix inequalities and majorizations around Hermite–Hadamard’s inequality

On Orthogonal Polynomials With Respect to a Positive Definite Matrix of Measures

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2 results

Matrix inequalities and majorizations around Hermite–Hadamard’s inequality

On Orthogonal Polynomials With Respect to a Positive Definite Matrix of Measures