Spaces Of Analytic Functions Research Articles

Knots and Their Related $q$-Series

We discuss a matrix of periodic holomorphic functions in the upper and lower half-plane which can be obtained from a factorization of an Andersen-Kashaev state integral of a knot complement with remarkable analytic and asymptotic properties that defines a ${\rm PSL}_2({\mathbb Z})$-cocycle on the space of matrix-valued piecewise analytic functions on the real numbers. We identify the corresponding cocycle with the one coming from the Kashaev invariant of a knot (and its matrix-valued extension) via the refined quantum modularity conjecture of [arXiv:2111.06645] and also relate the matrix-valued invariant with the 3D-index of Dimofte-Gaiotto-Gukov. The cocycle also has an analytic extendability property that leads to the notion of a matrix-valued holomorphic quantum modular form. This is a tale of several independent discoveries, both empirical and theoretical, all illustrated by the three simplest hyperbolic knots.

Hilbert Space Decomposition Properties of Complex Functions and Their Applications

We analyzed the classical problem of decomposing the Hilbert space of holomorphic functions, especially their splitting into the product or sum of domain-separated components. For the Bergman space of analytical functions, we obtained a special decomposition satisfying the assigned growth degree properties. Concerning a general Hilbert space of analytical functions on a connected domain, we studied its α-invariant decomposition and related ergodic consequences. As an interesting consequence, we obtained the decomposition theorem for an ergodic α-mapping on the Bergman space of holomorphic functions.

On the extension of the Voronin universality theorem for the Riemann zeta-function

In the paper, we prove a general universality theorem for the Riemann zeta-function ζ (s) on approximation of a class of analytic functions by generalized shifts ζ (s + ia(τ)). Here a(τ) is a real-valued continuous increasing to +∞ function, uniformly distributed modulo 1 and such that |ζ(σ + ia(τ) + it)|2, for σ > 1/2, has a traditional mean estimate for every t ∈ ℝ. For example, the function tv (τ), v > 0, where t(τ) is the Gram function, satisfies the hypotheses of the proved theorem. For the proof, the method of weak convergence of probabilistic measures in the space of analytic functions is developed.

Hilbert-type operator induced by radial weight on Hardy spaces

We consider the Hilbert-type operator defined by Hω(f)(z)=∫01f(t)1z∫0zBtω(u)duω(t)dt,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} H_{\\omega }(f)(z)=\\int _0^1 f(t)\\left( \\frac{1}{z}\\int _0^z B^{\\omega }_t(u)\\,du\\right) \\,\\omega (t)dt, \\end{aligned}$$\\end{document}where {Bζω}ζ∈D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{B^{\\omega }_\\zeta \\}_{\\zeta \\in \\mathbb {D}}$$\\end{document} are the reproducing kernels of the Bergman space Aω2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^2_\\omega $$\\end{document} induced by a radial weight ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} in the unit disc D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {D}$$\\end{document}. We prove that Hω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{\\omega }$$\\end{document} is bounded on the Hardy space Hp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^p$$\\end{document}, 1<p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<\\infty $$\\end{document}, if and only if sup0≤r<1ω^(r)ω^1+r2<∞,(†)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sup _{0\\le r<1} \\frac{\\widehat{\\omega }(r)}{\\widehat{\\omega }\\left( \\frac{1+r}{2}\\right) }<\\infty , (\\dag )\\end{aligned}$$\\end{document}and sup0<r<1∫0r1ω^(t)pdt1p∫r1ω^(t)1-tp′dt1p′<∞,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sup \\limits _{0<r<1}\\left( \\int _0^r \\frac{1}{\\widehat{\\omega }(t)^p} dt\\right) ^{\\frac{1}{p}} \\left( \\int _r^1 \\left( \\frac{\\widehat{\\omega }(t)}{1-t}\\right) ^{p'}\\,dt\\right) ^{\\frac{1}{p'}} <\\infty , \\end{aligned}$$\\end{document}where ω^(r)=∫r1ω(s)ds\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widehat{\\omega }(r)=\\int _r^1 \\omega (s)\\,ds$$\\end{document}. We also prove that Hω:H1→H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_\\omega : H^1\\rightarrow H^1$$\\end{document} is bounded if and only if (†\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dag $$\\end{document}) holds and supr∈[0,1)ω^(r)1-r∫0rdsω^(s)<∞.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sup \\limits _{r \\in [0,1)} \\frac{\\widehat{\\omega }(r)}{1-r} \\left( \\int _0^r \\frac{ds}{\\widehat{\\omega }(s)}\\right) <\\infty . \\end{aligned}$$\\end{document}As for the case p=∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=\\infty $$\\end{document}, Hω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_\\omega $$\\end{document} is bounded from H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^\\infty $$\\end{document} to BMOA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathord \ extrm{BMOA}$$\\end{document}, or to the Bloch space, if and only if (†\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dag $$\\end{document}) holds. In addition, we prove that there does not exist radial weights ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} such that Hω:Hp→Hp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{\\omega }: H^p \\rightarrow H^p $$\\end{document}, 1≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p<\\infty $$\\end{document}, is compact and we consider the action of Hω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{\\omega }$$\\end{document} on some spaces of analytic functions closely related to Hardy spaces.

Metrical Boundedness and Compactness of a New Operator between Some Spaces of Analytic Functions

The metrical boundedness and metrical compactness of a new operator from the weighted Bergman-Orlicz spaces to the weighted-type spaces and little weighted-type spaces of analytic functions are characterized.

On a linear operator between weighted‐type spaces of analytic functions

We characterize the boundedness, compactness, and estimate essential norm of the polynomial differentiation composition operator between weighted‐type spaces of analytic functions, in terms of the operator symbols and weights.

Limit theorems for random Dirichlet series

We prove a functional limit theorem in a space of analytic functions for the random Dirichlet series D(α;z)=∑n≥2(logn)α(ηn+iθn)/nz, properly scaled and normalized, where (ηn,θn)n∈N is a sequence of independent copies of a centered R2-valued random vector (η,θ) with a finite second moment and α>−1/2 is a fixed real parameter. As a consequence, we show that the point processes of complex and real zeros of D(α;z) converge vaguely, thereby obtaining a universality result. In the real case, that is, when P{θ=0}=1, we also prove a law of the iterated logarithm for D(α;z), properly normalized, as z→(1/2)+.

Thomson decompositions of measures in the disk

We study the classical problem of identifying the structure of P 2 ( μ ) \mathcal {P}^2(\mu ) , the closure of analytic polynomials in the Lebesgue space L 2 ( μ ) L^2(\mu ) of a compactly supported Borel measure μ \mu living in the complex plane. In his influential work, Thomson [Ann. of Math. (2) 133 (1991), pp. 477–507] showed that the space decomposes into a full L 2 L^2 -space and other pieces which are essentially spaces of analytic functions on domains in the plane. For a family of measures μ \mu supported on the closed unit disk D ¯ \overline {\mathbb {D}} which have a part on the open disk D \mathbb {D} which is similar to the Lebesgue area measure, and a part on the unit circle T \mathbb {T} which is the restriction of the Lebesgue linear measure to a general measurable subset E E of T \mathbb {T} , we extend the ideas of Khrushchev and calculate the exact form of the Thomson decomposition of the space P 2 ( μ ) \mathcal {P}^2(\mu ) . It turns out that the space splits according to a certain natural decomposition of measurable subsets of T \mathbb {T} which we introduce. We highlight applications to the theory of the Cauchy integral operator and de Branges-Rovnyak spaces.

Automorphisms and generalized projections on spaces of analytic functions

We present complete classifications of automorphisms of two closed subalgebras of the space of bounded analytic functions on the open unit disc D, namely, the subalgebra of functions vanishing at the origin, and the subalgebra of functions whose first derivative vanishes at the origin. The later subalgebra is known as the Neil algebra. We also characterize generalized tri-circular projections on Hp(D) and Hp(D2), 1≤p≤∞, p≠2.

A comparative study for the spectral properties of Toeplitz and Hankel operators

In this introductory review, we study Hankel and Toeplitz operators considering them as acting on certain spaces of analytic functions, namely Hardy spaces and compare their spectral properties such as their compactness criteria. In contrast to Toeplitz operators, the symbol of a Hankel operator is not uniquely determined by the operator. We also connect Toeplitz operators with Fredholm operators and give some of the most beautiful properties of Toeplitz operators such as the essential spectrum of Toeplitz operator with continuous symbol and the index of Toeplitz operator introducing Fredholm operators firstly.

Composition-differentiation operators acting on certain Hilbert spaces of analytic functions

We study composition-differentiation operators acting on the Bergman and Dirichlet space of the open unit disk. We first characterize the compactness of composition-differentiation operator on weighted Bergman spaces. We shall then prove that for an analytic self-map $\varphi$ on the open unit disk $\mathbb{D}$, the induced composition-differentiation operator is bounded with dense range if and only if $\varphi$ is univalent and the polynomials are dense in the Bergman space on $\Omega:=\varphi(\mathbb{D})$.

Approximation of smooth functionals using deep ReLU networks

In recent years, deep neural networks have been employed to approximate nonlinear continuous functionals F defined on Lp([−1,1]s) for 1≤p≤∞. However, the existing theoretical analysis in the literature either is unsatisfactory due to the poor approximation results, or does not apply to the rectified linear unit (ReLU) activation function. This paper aims to investigate the approximation power of functional deep ReLU networks in two settings: F is continuous with restrictions on the modulus of continuity, and F has higher order Fréchet derivatives. A novel functional network structure is proposed to extract features of higher order smoothness harbored by the target functional F. Quantitative rates of approximation in terms of the depth, width and total number of weights of neural networks are derived for both settings. We give logarithmic rates when measuring the approximation error on the unit ball of a Hölder space. In addition, we establish nearly polynomial rates (i.e., rates of the form exp−a(logM)b with a>0,0<b<1) when measuring the approximation error on a space of analytic functions.

Pointwise multipliers between spaces of analytic functions

A Banach space X of analytic function in , the unit disc in , is said to be admissible if it contains the polynomials and convergence in X implies uniform convergence in compact subsets of . If X and Y are two admissible Banach spaces of analytic functions in and g is a holomorphic function in , g is said to be a multiplier from X to Y if g ·f ∈ Y for every f ∈ X. The space of all multipliers from X to Y is denoted M (X, Y), and M (X) will stand for M (X, X). The closed graph theorem shows that if g ∈ M (X, Y) then the multiplication operator M g, defined by Mg (f) = g · f, is a bounded operator from X into Y. It is known that M (X) ⊂ H ∞ and that if g ∈ M (X), then ∥g∥H ∞ ≤ ||Mg||. Clearly, this implies that M (X, Y) ⊂ H ∞ if Y ⊂ X. If Y ⊄ X, the inclusion M (X, Y) ⊂ H ∞ may not be true. In this paper we start presenting a number of conditions on the spaces X and Y which imply that the inclusion M (X, Y) ⊂ H ∞ holds. Next, we concentrate our attention on multipliers acting an BMOA and some related spaces, namely, the Qs-spaces (0<s<∞).

Real symmetric, unitary, and complex symmetric weighted composition operators on Bergman spaces of polydisk

In this paper, we study weighted composition operators on Bergman spaces of analytic functions which are square integrable on polydisk. We develop the study in full generality, meaning that the corresponding weighted composition operators are not assumed to be bounded. The properties of weighted composition operators such as real symmetry, unitariness, complex symmetry, are characterized fully in simple algebraic terms, involving their symbols. As it turns out, a weighted composition operator having a symmetric structure must be bounded. We also obtain the interesting consequence that real symmetric weighted composition operators are complex symmetric corresponding an adapted and highly relevant conjugation.

Topological transitivity of translation operators in a non-separable Hilbert space

We consider a Hilbert space of entire analytic functions on a non-separable Hilbert space, associated with a non-separable Fock space. We show that under some conditions operators, like the differentiation operators and translation operators, are topologically transitive in this space.

Essential norm of intrinsic operators from Banach spaces of analytic functions into weighted-type spaces

Essential norm of intrinsic operators from Banach spaces of analytic functions into weighted-type spaces

On the Approximation by Mellin Transform of the Riemann Zeta-Function

This paper is devoted to the approximation of a certain class of analytic functions by shifts Z(s+iτ), τ∈R, of the modified Mellin transform Z(s) of the square of the Riemann zeta-function ζ(1/2+it). More precisely, we prove the existence of a closed non-empty set F such that there are infinitely many shifts Z(s+iτ), which approximate a given analytic function from F with a given accuracy. In the proof, the weak convergence of measures in the space of analytic functions is applied. Then, the set F coincides with the support of a limit measure.

Generalized Universality for Compositions of the Riemann Zeta-Function in Short Intervals

In the paper, the approximation of analytic functions on compact sets of the strip {s=σ+it∈C∣1/2&lt;σ&lt;1} by shifts F(ζ(s+iu1(τ)),…,ζ(s+iur(τ))), where ζ(s) is the Riemann zeta-function, u1,…,ur are certain differentiable increasing functions, and F is a certain continuous operator in the space of analytic functions, is considered. It is obtained that the set of the above shifts in the interval [T,T+H] with H=o(T), T→∞, has a positive lower density. Additionally, the positivity of a density with a certain exceptional condition is discussed. Examples of considered operators F are given.

Norms of a Product of Integral and Composition Operators between Some Bloch-Type Spaces

We present some formulas for the norm, as well as the essential norm, of a product of composition and an integral operator between some Bloch-type spaces of analytic functions on the unit ball, in terms of given symbols and weights.

On the Discrete Approximation by the Mellin Transform of the Riemann Zeta-Function

In the paper, it is obtained that there are infinite discrete shifts Ξ(s+ikh), h&gt;0, k∈N0 of the Mellin transform Ξ(s) of the square of the Riemann zeta-function, approximating a certain class of analytic functions. For the proof, a probabilistic approach based on weak convergence of probability measures in the space of analytic functions is applied.