These are some notes written while visiting IISc Bangalore. This was a discussion session with Apoorva Khare and Sujit Damase. An important reference to see these things in more detail is Apoorva’s survey [1].
Take a function \(f: I \rightarrow \mathbb{R}\), for \(I=\) any domain in \(\mathbb{R}\). Let \((a_{ij}) \in \mathbb{P}_{n}(I)\), where \(\mathbb{P}_{n}(I)\) is the set of \(n \times n\) positive semidefinite matrices with entries in \(I\).
We want to characterize \(f\) such that for all \((a_{ij}) \in \mathbb{P}_{n}(I)\), \((f(a_{ij})) \in \mathbb{P}_{n}(\mathbb{R})\).
Let’s call this family \(\mathcal{F}_{\infty}\).
\(f=1\) obviously works, and \(f(x)=x\) also works. The class of \(f\) that work is closed under addition, scaling by positive reals and pointwise limits. So it is a closed convex cone. It is also closed under product because of the Schur product theorem (1911).
Schur showed that matrices preserve PSD condition under Hadamard products (which should be named after Schur actually). So this easily implies being closed under products.
These above remarks can be collected to this theorem.
(1925) Polya-Szegos book exercise
The positivity preservers \(\mathcal{F}_{\infty}\) contain the closure of all polynomials and therefore contains all power series with positive coefficients that are convergent on \(I\).
Are there any other functions? What are they?
(Schoenberg, 1942)
(Rudin, 1959)
(Rudin, 1963)
Now, let’s go back to \(I=[-1,1]\). In Rudin’s paper, he also noticed that the end points might introduce some new functions. For example \(x,x^{3},x^{5},\dots\) converge to \(\mathbf{1}_{1}- \mathbf{1}_{-1}\) where \(\mathbf{1}\) is the indicator function. Let \(\varphi_{-1}(x)= \mathbf{1}_{1} - \mathbf{1}_{-1}\).
Similarly, the even powers \(1,x^{2},x^{4},\dots\) converge to \(\mathbf{1}_{\{-1,1\}} = \varphi_{-2}(x)\). Then, Rudin asked if there any others?
(Christensen-Ressel, 1970s)
\(I=[-1,1]\) for any \(f\). Then \(\mathcal{F}_{\infty}([-1,1]) = \sum_{k \geq -2}^{}a_{k} \varphi_{k}(x)\) and \(a_{k} \geq 0\) and for \(k \geq 0\), \(\varphi_{k}(x) = x^{k}\).
There is also a related work for the closed disc for which you must see Apoorva’s note.
Define \[\begin{equation} \mathcal{F}_{n}(I) = \{ \text{ positivity preservers for $n \times n$ matrices }\} \end{equation}\]
Open question
What is \(\mathcal{F}_{3}((0,\infty])\)?
Which polynomials work for \(\mathcal{F}_{n}(I)\)? This is known. Apoorva has a result where they show that \(\mathcal{P}_{n}(I)\) have polynomials with non-negative coefficients. This is about his work with Terry Tao.
There is also a theorem of Loeuvner (1967).
Let \(f \in \mathcal{F}_{n}((0,\infty)) \implies f \in C^{(n-3)}((0,\infty))\) and \(f,f^{1},\dots,f^{(n-3)} \geq 0\).
Positivity preservers are used as cutoffs (regularizers) of covariance matrices in statistics. Apoorva explained this using how this is connected to modeling weather.
Let \((X,d)\) be a finite metric space. All the data of \((X,d)\) is inside a distance metric graph.
\[\begin{equation} D_{X} = (d_{X}(x_{i},x_{j}))_{i,j=0}^{n} \end{equation}\]
Frechet (1010)
There is an isometric embedding \((X_{n},d) \rightarrow (\mathbb{R}^{n}, \| \cdot\|_{\infty})\). Witsenhausen sharpened it to \(\mathbb{R}^{n-1}\) for \(n \geq 3\).
When does \((X_{n},d) \rightarrow (\mathbb{R}^{m} , \|\cdot\|_{2})\) isometrically? Clearly \(m=n\) should be sufficient. What is the minimum \(m\) given \(X_{n}\)?
(Schoenberg, 1935)
Construct \(\mathrm{CM}(X) = (d_{i0}^{2} + d_{j0}^{2} - d_{ij}^{2})_{i,j=1}^{n}\). Then \(X\) embeds in \(\mathbb{R}^{n}\) with \(\|\cdot\|_{2}\)-norm if and only if \(\mathrm{CM}(X)\) is PSD. Also, minimum dimension \(m = \mathrm{rank}\mathrm{CM}(X)\).
Let us prove the \(\implies\) implication. Observe that \[\begin{align} d_{10}^{2} + d_{j0}^{2} - d_{ij}^{2} & = \|x_{i}-x_{0}\|^{2} + \|x_{j}-x_{0}\|^{2} - \|(x_{i}-x_{0}) - (x_{j}- x_{0})\|^{2} \\ & = 2 \langle x_{i} - x_{0}, x_{j} - x_{0}\rangle. \end{align}\] This shows that \(\mathrm{CM}(X)\) is a Gramm matrix.
The other direction is to basically find \(n\) points such that \(\mathrm{CM}(X)\) is a Gramm matrix of those \(n\)-points and map \(X\) to those \(n\)-points.
In the same paper, Schoenberg also dealt with embedding \(X \rightarrow (S^{n}, \angle)\) for the \(n\)-dimensional sphere in \(\mathbb{R}^n\). This is an application of positivity preservers.
(Schoenberg 1935)
There exists an isometric embedding \(X \rightarrow (S^{n}, \angle)\) for the \(n\)-dimensional sphere if and only if
(Schoenberg) What are all the positive definite functions on \(S^{n-1}\)?
He answered this in his 1942. This is the same as asking which functions \(f \circ \cos^{-1}: [-1,1] \rightarrow \mathbb{R}\) send correlation matrices to correlation matrices. This is answered as \[\begin{equation} f = \sum_{k=0}^{\infty} a_{k} C_{k}^{(\cdot)} (x), \text{ with }a_{k} \geq 0, \end{equation}\] and \(C_{k}^{(\cdot)}(x)\) are some Gegenbauer polynomials.
(Schoenberg)
What about \(S^{\infty}\)? This is the set of \(l^{2}\)-sequences with \(l^{2}\)-norm 1.
This is answered in Schoenberg’s 1942 paper. The answer is \[\begin{equation} \bigcap_{n \geq 3}\{ \sum_{k \geq 0}^{} a_{k} C_{k}^{(\frac{n-3}{2})}(x)\} = \{ \sum_{k \geq 0}^{}a_{k} x^{k}\}. \end{equation}\] Here \(a_{k}\) are non-negative. This is also the same as answering the question of Polya-Szego.
A p.d. kernel is a functon \(K: X \times X \rightarrow \mathbb{C}\) on a finite set if the matrix \(K(x_{i},x_{j})\) is psd. For a group \(G\), the function \(\varphi: G \rightarrow \mathbb{C}\) is psd if \(\varphi(g_{i}^{-1} g_{k})\) is psd.
For a function \(K:G \rightarrow G \rightarrow \mathbb{C}\), \(K(x,y)\) is \(G\)-invariant if \(K(gx,gy)= K(x,y)\) for all \(g \in G\). Then, \(\psi(x)=K(x,e)\) is just a one-variable function on \(G\) that determines the kernel \(K\).
We conisder a two-point homogeneous space \((X,d)\), that is a metric \(G\)-space such that \[\begin{equation} d(x_{1},y_{1})=d(x_{2},y_{2}) \implies \exists g \in G, gx_{1} = y_{1}, g x_{2}=y_{2}. \end{equation}\]
\(K: X \times X \rightarrow \mathbb{C}\) be a function on some \(G\)-space \(X\), for eg. \(X=S^{n-1} \subseteq \mathbb{R}^{n}\), \(G = \mathrm{SO}(n)\). Then because of \(\mathrm{SO}(n-1)\)-invriance, \(K\) becomes a function of \(\langle x,y \rangle\). Hence, \(K = \varphi(\langle x,y\rangle)\) for a one-variable function \(\varphi: [-1,1] \rightarrow \mathbb{C}\). Then, \(K\) is a positive-definite kernel if and only if \(\varphi\) is a positivity preserver.
(Bochner)
Let \(G\) be a locally compact Abelian group. Then \(f: G \rightarrow \mathbb{C}\) is a positive definite function if and only if there exists some \(\mu \in M(\widehat{G})\)=finite, positive measures on \(\widehat{G}\) such that \[\begin{equation} f = \int_{\widehat{G}}^{} \chi(t) \, \mathrm{d}\mu (\chi) \end{equation}\]
For example, if \(G = S^{1}\), \(\widehat{G} = \mathbb{Z}\) then the positive definite functions on \(G\) are \[\begin{equation} f(t) = \sum_{n \in \mathbb{Z}}^{} e^{i nt} \mu_{n}, \end{equation}\] for some \(l^{1}\)-sequence \(\{\mu_{n}\}_{n \in \mathbb{Z}}\). These essentially look like power series with non-negative coefficients.
For a general sphere, \[\begin{equation} L^{2}(X) = \bigoplus_{j} V_{j}. \end{equation}\] This is a subspace of \(L^{2}(G)\), the \(\mathrm{SO}(n-1)\)-invariant functions of \(\mathrm{SO}(n)\). The Gegenbauer polynomials form an orthonormal basis for \(V_{k}\). Hence, we can write a version of Bochner’s theorem as \[\begin{equation} f(\langle x,y\rangle) = K(x,y) = \sum_{k=0}^{\infty} a_{k} C_{k}^{\frac{n-3}{2}}, \end{equation}\] where \[\begin{equation} a_{k} = \int_{S^{n-1}}^{} K(x,y) C_{k}^{\frac{n-3}{2}}(x,y) \, \mathrm{d}\nu (x). \end{equation}\]
This page was updated on January 30, 2026.
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