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Register a new userTotally positive kernels, Polya frequency functions, and their transforms
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The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown that such a composition operator maps the set of totally non-negative kernels to itself if and only if the function is constant or linear, or just linear if it preserves total positivity. Symmetric kernels are also discussed, with a similar outcome. These classification results are a byproduct of two matrix-completion results and the second theme: an extension of A.M. Whitneys density theorem from finite domains to subsets of the real line. This extension is derived via a discrete convolution with modulated Gaussian kernels. The third theme consists of analyzing, with tools from harmonic analysis, the preservers of several families of totally non-negative and totally positive kernels with additional structure: continuous Hankel kernels on an interval, Polya frequency functions, and Polya frequency sequences. The rigid structure of post-composition transforms of totally positive kernels acting on infinite sets is obtained by combining several specialized situations settled in our present and earlier works.
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We prove the converse to a result of Karlin [Trans. AMS 1964], and also strengthen his result and two results of Schoenberg [Ann. of Math. 1955]. One of the latter results concerns zeros of Laplace transforms of multiply positive functions. The other results study which powers $alpha$ of two specific kernels are totally non-negative of order $pgeq 2$ (denoted TN$_p$); both authors showed this happens for $alphageq p-2$, and Schoenberg proved that it does not for $alpha<p-2$. We show more strongly that for every $p times p$ submatrix of either kernel, up to a shift, its $alpha$th power is totally positive of order $p$ (TP$_p$) for every $alpha > p-2$, and is not TN$_p$ for every non-integer $alphain(0,p-2)$. In particular, these results reveal critical exponent phenomena in total positivity. We also prove the converse to a 1968 result of Karlin, revealing yet another critical exponent phenomenon - for Laplace transforms of all Polya Frequency (PF) functions. We further classify the powers preserving all TN$_p$ Hankel kernels on intervals, and isolate individual kernels encoding these powers. We then transfer results on preservers by Polya-Szego (1925), Loewner/Horn (1969), and Khare-Tao (in press), from positive matrices to Hankel TN$_p$ kernels. Another application constructs individual matrices encoding the Loewner convex powers. This complements Jains results (2020) for Loewner positivity, which we strengthen to total positivity. Remarkably, these (strengthened) results of Jain, those of Schoenberg and Karlin, the latters converse, and the above Hankel kernels all arise from a single symmetric rank-two kernel and its powers: $max(1+xy,0)$. We also provide a novel characterization of PF functions and sequences of order $pgeq 3$, following Schoenbergs 1951 result for $p=2$. We correct a small gap in his paper, in the classification of discontinuous PF functions.
The main purpose of our paper is a new approach to design of algorithms of Kaczmarz type in the framework of operators in Hilbert space. Our applications include a diverse list of optimization problems, new Karhunen-Lo`eve transforms, and Principal Component Analysis (PCA) for digital images. A key feature of our algorithms is our use of recursive systems of projection operators. Specifically, we apply our recursive projection algorithms for new computations of PCA probabilities and of variance data. For this we also make use of specific reproducing kernel Hilbert spaces, factorization for kernels, and finite-dimensional approximations. Our projection algorithms are designed with view to maximum likelihood solutions, minimization of cost problems, identification of principal components, and data-dimension reduction.
We classify functions $f:(a,b)rightarrow mathbb{R}$ which satisfy the inequality $$operatorname{tr} f(A)+f(C)geq operatorname{tr} f(B)+f(D)$$ when $Aleq Bleq C$ are self-adjoint matrices, $D= A+C-B$, the so-called trace minmax functions. (Here $Aleq B$ if $B-A$ is positive semidefinite, and $f$ is evaluated via the functional calculus.) A function is trace minmax if and only if its derivative analytically continues to a self map of the upper half plane. The negative exponential of a trace minmax function $g=e^{-f}$ satisfies the inequality $$det g(A) det g(C)leq det g(B) det g(D)$$ for $A, B, C, D$ as above. We call such functions determinant isoperimetric. We show that determinant isoperimetric functions are in the radical of the the Laguerre-Polya class. We derive an integral representation for such functions which is essentially a continuous version of the Hadamard factorization for functions in the the Laguerre-Polya class. We apply our results to give some equivalent formulations of the Riemann hypothesis.
Given scalars $a_n ( eq 0)$ and $b_n$, $n geq 0$, the tridiagonal kernel or band kernel with bandwidth $1$ is the positive definite kernel $k$ on the open unit disc $mathbb{D}$ defined by [ k(z, w) = sum_{n=0}^infty Big((a_n + b_n z)z^nBig) Big((bar{a}_n + bar{b}_n bar{w}) bar{w}^n Big) qquad (z, w in mathbb{D}). ] This defines a reproducing kernel Hilbert space $mathcal{H}_k$ (known as tridiagonal space) of analytic functions on $mathbb{D}$ with ${(a_n + b_nz) z^n}_{n=0}^infty$ as an orthonormal basis. We consider shift operators $M_z$ on $mathcal{H}_k$ and prove that $M_z$ is left-invertible if and only if ${|{a_n}/{a_{n+1}}|}_{ngeq 0}$ is bounded away from zero. We find that, unlike the case of weighted shifts, Shimorins models for left-invertible operators fail to bring to the foreground the tridiagonal structure of shifts. In fact, the tridiagonal structure of a kernel $k$, as above, is preserved under Shimorin model if and only if $b_0=0$ or that $M_z$ is a weighted shift. We prove concrete classification results concerning invariance of tridiagonality of kernels, Shimorin models, and positive operators. We also develop a computational approach to Aluthge transforms of shifts. Curiously, in contrast to direct kernel space techniques, often Shimorin models fails to yield tridiagonal Aluthge transforms of shifts defined on tridiagonal spaces.
We establish a duality for two factorization questions, one for general positive definite (p.d) kernels $K$, and the other for Gaussian processes, say $V$. The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel $K$, presented as a covariance kernel for a Gaussian process $V$. We then give an explicit duality for these two seemingly different notions of factorization, for p.d. kernel $K$, vs for Gaussian process $V$. Our result is in the form of an explicit correspondence. It states that the analytic data which determine the variety of factorizations for $K$ is the exact same as that which yield factorizations for $V$. Examples and applications are included: point-processes, sampling schemes, constructive discretization, graph-Laplacians, and boundary-value problems.
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