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Mainak Bhowmik

Publications and source records attributed to Mainak Bhowmik.

11 recordsLinked to original sources
arXiv2026

Holomorphic Interpolation of Multivariate Completely Monotone Functions

The integral representation of completely monotone functions of several real variables as Laplace or Stieltjes-Fantappi\'e transforms of positive measures opens a Hilbert space path toward their finite-point interpolation by simpler functions. We combine, within a non-commutative Radon transform framework, the matrix pencil realization of the positive semi-definite Hankel kernel associated with the sampling of a completely monotone function with Weyl's operational calculus and Fantappi\`e's analytic calculus. The interpolation is achieved by finitely determined entire or rational functions, respectively, which are directionally completely monotone. In our relaxation scheme, the original positive measure is approximated by a sequence of specific Wigner distributions, which can also be regarded as analytic functionals. Throughout the interpolation process, tight bounds are enforced on the modulus or the real part of the holomorphic extension to the underlying tube domain.

math.FA↗
arXiv2026

A family of $2 \times 2$ tetrablock contractions with non-commuting fundamental operators

In this note, we provide a family of $2\times 2$ tetrablock contractions that have tetrablock isometric dilation, but the corresponding fundamental operators do not commute. This answers a question raised by Bhattacharyya [Indiana Univ. Math. J. 2014] about the necessity of commuting fundamental operators for a tetrablock contraction to have rational dilation in a negative direction.

math.FA↗
arXiv2025

The multivariate Herglotz-Nevanlinna class: Rational approximation

We return to Takagi's variational principle, generalized after forty years to two complex variables by Pfister. Both isolating some extremal rational functions associated to a bounded holomorphic function in the unit disk, respectively the bidisk. The rational inner functions arising from the Takagi-Pfister skew eigenvectors lead to a Pade type approximation scheme. For these rational functions, we prove a Montessus de Ballore type convergence theorem, on the polydisk in any complex dimension. On the natural and more restrictive class of Agler holomorphic functions with non-negative real part, we show that Cayley rational inner functions match any finite section of the Taylor expansion at a prescribed point. We derive from the Hilbert space proof that the finite section coefficient set of Taylor series of the Agler functions in the Herglotz-Nevanlinna setting is semi-algbraic. The pole distribution of the Takagi-Pfister interpolation sequence is identified as a main open question on the subject.

math.CV↗
arXiv2025

The Multivariate Herglotz-Nevanlinna Class: Superresolution

Bounded holomorphic interpolation problems associated to finitely many data have, in general, distinct solutions. Uniqueness arises only in some convex extreme configurations. Rational inner functions in a polydisk are the best understood examples in this sense. We analyze the continuity of global solutions as functions of the finite interpolation data in neighbourhoods of elements distinguished by this uniqueness property. Our study covers rational inner or Cayley rational inner functions in the polydisk and automorphisms of the Euclidean ball. The proof of the main superresolution result is derived from optimization theory techniques and volume estimates of sublevel sets of real polynomials, both emerging from Markov's multivariable moment problem.

math.FA↗
arXiv2024

Herglotz representation for operator-valued function on a set associated with test functions

The Herglotz representation theorem for holomorphic functions with non-negative real part is a fundamental result in the theory of holomorphic functions. In this paper, we reinterpret the Herglotz representation in the context of modern techniques, specifically realization formula. This reinterpretation is then extended to operator-valued functions on arbitrary sets, in association with a collection of test functions.

math.FA↗
arXiv2024

Hankel operators and Projective Hilbert modules on quotients of bounded symmetric domains

Consider a bounded symmetric domain $\Omega$ with a finite pseudo-reflection group acting on it as a subgroup of the group of automorphisms. This gives rise to quotient domains by means of basic polynomials $\theta$ which by virtue of being proper maps map the \v Silov boundary of $\Omega$ to the \v Silov boundary of $\theta(\Omega)$. Thus, the natural measure on the \v Silov boundary of $\Omega$ can be pushed forward. This gives rise to Hardy spaces on the quotient domain. The study of Hankel operators on the Hardy spaces of the quotient domains is introduced. The use of the weak product space shows that an analogue of Hartman's theorem holds for the small Hankel operator. Nehari's theorem fails for the big Hankel operator and this has the consequence that when the domain $\Omega$ is the polydisc $\mathbb D^d$, the {\em Hardy space} is not a projective object in the category of all Hilbert modules over the algebra $\mathcal A (\theta(\mathbb D^d))$ of functions which are holomorphic in the quotient domain and continuous on the closure $\overline {\theta(\mathbb D^d)}$. It is not a projective object in the category of cramped Hilbert modules either. Indeed, no projective object is known in these two categories. On the other hand, every normal Hilbert module over the algebra of continuous functions on the \v Silov boundary, treated as a Hilbert module over the algebra $\mathcal A (\theta(\mathbb D^d))$, is projective.

math.FA↗
arXiv2024

The Bishop-Phelps-Bollobas property for certain Banach spaces

Let $X$ be a complex Banach space. We prove that if $L$ is an extremally disconnected compact Hausdorff topological space, then the pair $(X, C(L))$ satisfies the Bishop-Phelps-Bollob\'as property (BPBp for short). As a byproduct, we obtain the BPBp for the pair $(X, L^\infty(\nu))$ for any measure $\nu$. In particular, this settles an unresolved question regarding the BPBp for the pair $(L^\infty(\mu), L^\infty(\nu) )$ for any two measures $\mu$ and $\nu$. Finally, we show that $(X,H^\infty(\Omega)$ has the BPBp when $\Omega$ is a multi-connected planar domain bounded by finitely many disjoint analytic simple closed curves.

math.FA↗
arXiv2023

Herglotz's representation and Caratheodory's approximation

Herglotz's representation of holomorphic functions with positive real part and Carath\'eodory's theorem on approximation by inner functions are two well-known classical results in the theory of holomorphic functions on the unit disc. We show that they are equivalent. On a multi-connected domain $\Omega$, a version of Heglotz's representation is known. Carath\'eodory's approximation was not known. We formulate and prove it and then show that it is equivalent to the known form of Herglotz's representation. Additionally, it also enables us to prove a new Heglotz's representation in the style of Koranyi and Pukanszky. Of particular interest is the fact that the scaling technique of the disc is replaced by Carath\'eodory's approximation theorem while proving this new form of Herglotz's representation. Carath\'eodory's approximation theorem is also proved for matrix-valued functions on a multi-connected domain.

math.FA↗
arXiv2022

Function theory on quotient domains related to the polydisc

Inner functions are the backbone of holomorphic function theory. This paper studies the inner functions on quotient domains of the open unit polydisc, $\bD^d$, arising from the group action of finite pseudo-reflection groups. Such quotient domains are known to be biholomorphic to the proper image $\theta(\bD^d)$ of $\bD^d$ under certain polynomial maps $\theta: \bD^d \to \theta(\bD^d)$. The main contributions of this paper are as follows: 1) We show that the closed algebra generated by inner functions on $\theta(\bD^d)$ forms a proper subalgebra of $H^\infty(\theta(\bD^d))$, the algebra of bounded holomorphic functions on $\theta(\bD^d)$. 2) The set of all rational inner functions on $\theta(\bD^d)$ is shown to be dense in the norm-unit ball of $H^\infty(\theta(\bD^d))$ with respect to the uniform compact-open topology, thereby proving the Carath\'eodory approximation result. 3) As an application of the Carath\'eodory approximation theorem, we approximate holomorphic functions on $\theta(\bD^d)$ that are continuous in the closure of ${\theta(\bD^d)}$ by convex combinations of rational inner functions in the $L^2 $-norm, thereby obtaining a version of the Fisher's theorem. 4) Given the two approximation results above, establishing a structure for rational inner functions is essential. We have identified the structure of rational inner functions on $\theta(\mathbb{D}^d)$. 5) The Carath\'eodory approximation for operator-valued functions is also discussed.

math.FA↗
arXiv2022

A note on the dilation of a certain family of tetrablock contractions

We find an explicit tetrablock isometric dilation for every member $(A_\alpha, B, P)$ of a family of tetrablock contractions indexed by a parameter $\alpha$ in the closed unit disc (only the first operator of the tetrablock contraction depends on the parameter). The dilation space is the same for any member of the family. Explicit dilation for the adjoint tetrablock contraction $(A_\alpha^*, B^*, P^*)$ for every member of the family mentioned above is constructed as well. This example is important because it has been claimed in the literature that this example does not have a dilation. Taking cue from this construction and using Toeplitz operators on $H^2_{\mathbb D}(\mathcal D_P)$, we obtain necessary and sufficient conditions for a tetrablock contraction to have a certain type of tetrablock isometric dilation.

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