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A Herglotz function is a holomorphic map from the open complex unit disk into the closed complex right halfplane. A classical Herglotz function has an integral representation against a positive measure on the unit circle. We prove a free analytic analogue of the Herglotz representation and describe how our representations specialize to the free probabilistic case. We also show that the set of representable Herglotz functions arising from noncommutative conditional expectations must be closed in a natural topology.
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Studies of sparse representation of deterministic signals have been well developed. Amongst there exists one called adaptive Fourier decomposition (AFD) established through adaptive selections of the parameters defining a Takenaka-Malmquist system in one-complex variable. The AFD type algorithms give rise to sparse representations of signals of finite energy. The multivariate generalization of AFD is one called pre-orthogonal AFD (POAFD), the latter being established with the context Hilbert space possessing a dictionary. The purpose of the present study is to generalize both AFD and POAFD to random signals. We work on two types of random signals. One is those expressible as the sum of a deterministic signal with an error term such as a white noise; and the other is, in general, as mixture of several classes of random signals obeying certain distributive law. In the first part of the paper we develop an AFD type sparse representation for one-dimensional random signals by making use analysis of one complex variable. In the second part, without complex analysis, we treat multivariate random signals in the context of stochastic Hilbert space with a dictionary. Like in the deterministic signal case the established random sparse representations are powerful tools in practical signal analysis.
For a wide family of even kernels ${varphi_u, uin I}$, we describe discrete sets $Lambda$ such that every bandlimited signal $f$ can be reconstructed from the space-time samples ${(fastvarphi_u)(lambda), lambdainLambda, uin I}$.
We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to $1/f$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modified to produce inner functions.
Let $x_0$ be a self-adjoint random variable and $c_t$ be a free circular Brownian motion, freely independent from $x_0$. We use the Hamilton--Jacobi method to compute the Brown measure $rho_t$ of $x_0+c_t$. The Brown measure is absolutely continuous with a density that is emph{constant along the vertical direction} in the support of $rho_t$. The support of the Brown measure of $x_0+c_t$ is related to the subordination function of the free additive convolution of $x_0+s_t$, where $s_t$ is the free semicircular Brownian motion, freely independent from $x_0$. Furthermore, the push-forward of $rho_t$ by a natural map is the law of $x_0+s_t$. Let $u$ be a unitary random variable and $b_t$ is the free multiplicative Brownian motion freely independent from $u$, we compute the Brown measure $mu_t$ of the free multiplicative Brownian motion $ub_t$, extending the recent work by Driver--Hall--Kemp. The measure is absolutely continuous with a density of the special form [frac{1}{r^2}w_t(theta)] in polar coordinates in its support. The support of $mu_t$ is related to the subordination function of the free multiplicative convolution of $uu_t$ where $u_t$ is the free unitary Brownian motion free independent from $u$. The push-forward of $mu_t$ by a natural map is the law of $uu_t$. In the special case that $u$ is Haar unitary, the Brown measure $mu_t$ follows the emph{annulus law}. The support of the Brown measure of $ub_t$ is an annulus with inner radius $e^{-t/2}$ and outer radius $e^{t/2}$. The density in polar coordinates is given by [frac{1}{2pi t}frac{1}{r^2}] in its support.
We address the following question: what can one say, for a tuple $(Y_1,dots,Y_d)$ of normal operators in a tracial operator algebra setting with prescribed sizes of the eigenspaces for each $Y_i$, about the sizes of the eigenspaces for any non-commutative polynomial $P(Y_1,dots,Y_d)$ in those operators? We show that for each polynomial $P$ there are unavoidable eigenspaces, which occur in $P(Y_1,dots,Y_d)$ for any $(Y_1,dots,Y_d)$ with the prescribed eigenspaces for the marginals. We will describe this minimal situation both in algebraic terms - where it is given by realizations via matrices over the free skew field and via rank calculations - and in analytic terms - where it is given by freely independent random variables with prescribed atoms in their distributions. The fact that the latter situation corresponds to this minimal situation allows to draw many new conclusions about atoms in polynomials of free variables. In particular, we give a complete description of atoms in the free commutator and the free anti-commutator. Furthermore, our results do not only apply to polynomials, but much more general also to non-commutative rational functions.
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