Related papers: Carath\'{e}odory interpolation on the non-commutat…
The Nevanlinna-Pick problem and the simplest case of the Carath\'eodory-Fej\'er problem on the spectral ball $\Om_3$ are reduced to interpolation problems on the symmetrized three-disc $\G_3.$
We consider a class of nonlinear parabolic equations \[ \dfrac{\partial}{\partial t} b(u)-\nabla \cdot (A(x,t,u,\nabla u))+H(x,t,\nabla u)=f , \] where $H$ is a nonlinear lower order term satisfied the Carath$\acute{e}$odory condition and…
A seminal result of Agler characterizes the so-called Schur-Agler class of functions on the polydisk in terms of a unitary colligation transfer function representation. We generalize this to the unit ball of the algebra of multipliers for a…
The higher order analogue of the classical Carath\'eodory-Julia theorem on boundary angular derivatives has been obtained in \cite{bk3}. Here we study boundary interpolation problems for Schur class functions (analytic and bounded by one in…
In this paper we improve the bounds for the Carath\'eodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower and upper bounds on algebraic varieties, $\mathbb{R}^n$, and…
The Schur (resp. Carath\'eodory) class consists of all the analytic functions $f$ on the unit disk with $|f|\le 1$ (resp. $\Re f>0$ and $f(0)=1$). The Schur parameters $\gamma_0,\gamma_1,\dots (|\gamma_j|\le 1)$ are known to parametrize the…
The Schur-Agler class consists of functions over a domain satisfying an appropriate von Neumann inequality. Originally defined over the polydisk, the idea has been extended to general domains in multivariable complex Euclidean space with…
This note is a companion paper to arXiv:1608.01630 [math.CA]. Here we generalize some of the geometric results of arXiv:1608.01630 [math.CA] to the case of a $3\times 3$ matrix function $A(x)\approx \mathrm{diag}\{1,f(x_1), g(x_1)\}$. More…
In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents (-\Delta)_{p(\cdot)}^{s(\cdot)}u(x)&=\lambda|u(x)|^{\alpha(x)-2}u(x)+…
In this paper, we show that the Carath\'{e}odory function $\varphi_{\scriptscriptstyle {Ne}}(z)=1+z-z^3/3$ maps the open unit disk $\mathbb{D}$ onto the interior of the nephroid, a $2$-cusped kidney-shaped curve, \begin{align*}…
Let $\Tei_{g,n}$ be the Teichm\"uller space of Riemann surfaces of genus $g$ with $n$ punctures. It is conjectured that the Teichm\"uller and Carath\'{e}odory metrics agree on a Teichm\"{u}ller disk if and only if all the zeros of the…
Carath\'eodory functions, i.e. functions analytic in the open upper half-plane and with a positive real part there, play an important role in operator theory, $1D$ system theory and in the study of de Branges-Rovnyak spaces. The Herglotz…
The nondegenerate Nevanlinna-Pick-Carath\'eodory-Fejer interpolation problem with finitely many interpolation conditions always has infinitely many solutions in a generalized Schur class $\cS_\kappa$ for every $\kappa\ge \kappa_{\rm min}$…
We describe a generalization of the notion of a Hilbert space model of a function $\varphi$ in the Schur-Agler class of the polydisc. This generalization is well adapted to the investigation of boundary behavior of $\varphi$ at a mild…
Characterization of Schur-class functions (analytic and bounded by one in modulus on the open unit disk) in terms of their Taylor coefficients at the origin is due to I. Schur. We present a boundary analog of this result: necessary and…
We define a class of pseudo-differential operators in a completely new way, which is called the abstract operators and expounded systematically the theory of abstract operators. By combining abstract operators with the Laplace transform, we…
We study the copolynomials of $n$ variables, i.e. $K$-linear mappings from the ring of polynomials $K[x_1,...,x_n]$ into the commutative ring $K$. We prove an existence and uniqueness theorem for a linear differential equation of infinite…
In this article, we establish a connection between Pick bodies and invariant functions. We demonstrate that an invariant function can be associated with any Pick body, which determines the solvability of a given Pick interpolation problem…
We consider the following convective Neumann systems:\begin{equation*}\left(\mathrm{S}\right)\qquad\left\{\begin{array}{ll}-\Delta_{p_1}u_1+\frac{|\nabla u_1|^{p_1}}{u_1+\delta_1}=f_1(x,u_1,u_2,\nabla u_1,\nabla u_2) & \text{in}\;\Omega,\\…
It is shown that for inhomogeneous Lie algebras $\frak{g}=\frak{s}\overrightarrow{\oplus}_{\Lambda}(\dim \Lambda)L_{1}$ satisfying the condition $\mathcal{N}(\frak{g})=1$, the only Casimir operator can be explicitly constructed from the…