Hardy Space Of Functions Research Articles

A note on Bohr’s theorem for Beurling integer systems

Given a sequence of frequencies {λn}n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{\\lambda _n\\}_{n\\ge 1}$$\\end{document}, a corresponding generalized Dirichlet series is of the form f(s)=∑n≥1ane-λns\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(s)=\\sum _{n\\ge 1}a_ne^{-\\lambda _ns}$$\\end{document}. We are interested in multiplicatively generated systems, where each number eλn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^{\\lambda _n}$$\\end{document} arises as a finite product of some given numbers {qn}n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{q_n\\}_{n\\ge 1}$$\\end{document}, 1<qn→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 < q_n \\rightarrow \\infty $$\\end{document}, referred to as Beurling primes. In the classical case, where λn=logn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _n = \\log n$$\\end{document}, Bohr’s theorem holds: if f converges somewhere and has an analytic extension which is bounded in a half-plane {ℜs>θ}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{\\Re s> \ heta \\}$$\\end{document}, then it actually converges uniformly in every half-plane {ℜs>θ+ε}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{\\Re s> \ heta +\\varepsilon \\}$$\\end{document}, ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon >0$$\\end{document}. We prove, under very mild conditions, that given a sequence of Beurling primes, a small perturbation yields another sequence of primes such that the corresponding Beurling integers satisfy Bohr’s condition, and therefore the theorem. Applying our technique in conjunction with a probabilistic method, we find a system of Beurling primes for which both Bohr’s theorem and the Riemann hypothesis are valid. This provides a counterexample to a conjecture of H. Helson concerning outer functions in Hardy spaces of generalized Dirichlet series.

Free outer functions in complete Pick spaces

Jury and Martin establish an analogue of the classical inner-outer factorization of Hardy space functions. They show that every function f f in a Hilbert function space with a normalized complete Pick reproducing kernel has a factorization of the type f = φ g f=\varphi g , where g g is cyclic, φ \varphi is a contractive multiplier, and ‖ f ‖ = ‖ g ‖ \|f\|=\|g\| . In this paper we show that if the cyclic factor is assumed to be what we call free outer, then the factors are essentially unique, and we give a characterization of the factors that is intrinsic to the space. That lets us compute examples. We also provide several applications of this factorization.

Integral means of derivatives of univalent functions in Hardy spaces

We show that the norm in the Hardy space H p H^p satisfies ( † ) ‖ f ‖ H p p ≍ ∫ 0 1 M q p ( r , f ′ ) ( 1 − r ) p ( 1 − 1 q ) d r + | f ( 0 ) | p \begin{equation} \|f\|_{H^p}^p\asymp \int _0^1M_q^p(r,f’)(1-r)^{p\left (1-\frac 1q\right )}\,dr+|f(0)|^p\tag {$\dagger $} \end{equation} for all univalent functions provided that either q ≥ 2 q\ge 2 or 2 p 2 + p &gt; q &gt; 2 \frac {2p}{2+p}&gt;q&gt;2 . This asymptotic was previously known in the cases 0 &gt; p ≤ q &gt; ∞ 0&gt;p\le q&gt;\infty and p 1 + p &gt; q &gt; p &gt; 2 + 2 157 \frac {p}{1+p}&gt;q&gt;p&gt;2+\frac {2}{157} by results due to Pommerenke [Math. Ann. 145 (1961/62), pp. 285–296], Baernstein, Girela and Peláez [Illinois J. Math. 48 (2004), pp. 837–859] and González and Peláez [J. Geom. Anal. 19 (2009), pp. 755–771]. It is also shown that ( † ) (\dagger ) is satisfied for all close-to-convex functions if 1 ≤ q &gt; ∞ 1\le q&gt;\infty . A counterpart of ( † ) (\dagger ) in the setting of weighted Bergman spaces is also briefly discussed.

Kakeya Inequalities by Maximal Functions in Hardy Spaces

In this paper, we will introduce and study several types of Kakeya inequalities by the maximal functions in Hardy spaces in ℝ n , n ≥ 2 , and we could obtain several inequalities associated with the Kakeya inequalities. We will show that M δ S α , β t f p ≲ p , n , φ , ε 1 / δ 5 n / r + 2 ε f p , when f x ∈ L p ℝ n and supp f ^ ξ ⊆ B 0 , 1 .

Univariate tight wavelet frames of minimal support

This work characterizes (dyadic homogeneous) wavelet frames for $$L^2({{\mathbb {R}}})$$ by means of spectral techniques. These techniques use decomposability properties of the frame operator in spectral representations associated with the dilation operator. The approach is closely related to usual Fourier domain fiberization techniques, dual Gramian analysis, and extension principles. Spectral formulas are used to determine all the tight wavelet frames for $$L^2({{\mathbb {R}}})$$ with a fixed finite number of generators of minimal support. The method associates wavelet frames of this type with certain inner operator-valued functions in Hardy spaces. The cases with one and two generators are completely solved.

On the Taylor Coefficients of Functions in the Hardy Space Over the Bidisc

In this paper we analyze the Taylor coefficients of functions in Hardy spaces on the bidisc. We present the two-dimensional versions of some classical Hardy and Paley inequalities and we find certain conditions on the Taylor coefficients for the converse of these inequalities to hold.

Boundary behavior of Hardy spaces on angular domains

The aim of this paper is to characterize the behavior of boundary limits of functions in Hardy space on angular domains. We investigate that the Cauchy integral of a function $$f\in L^{p}(\partial D_a)$$ is in $$H^{p}(D_a).$$ We also prove that the functions in $$H^{p}(D_a)$$ are the Cauchy integral of their non-tangential boundary limits. In addition, we establish the orthogonality of non-tangential boundary limits of functions in $$H^{p}(D_a)$$ and $$H^{q}(D_a)$$.

Fourier spectrum of Clifford Hp spaces on [formula omitted] for 1 ≤ p ≤ ∞

This article studies the Fourier spectrum characterization of functions in the Clifford algebra-valued Hardy spaces Hp(R+n+1),1≤p≤∞. Namely, for f∈Lp(Rn), Clifford algebra-valued, f is further the non-tangential boundary limit of some function in Hp(R+n+1), 1≤p≤∞, if and only if fˆ=χ+fˆ, where χ+(ξ_)=12(1+iξ_|ξ_|), the Fourier transformation and the above relation are suitably interpreted (for some cases in the distribution sense). These results further develop the relevant context of Alan McIntosh. As a particular case of our results, the vector-valued Clifford Hardy space functions are identical with the conjugate harmonic systems in the work of Stein and Weiss. The latter proved the corresponding singular integral version of the vector-valued cases for 1≤p<∞. We also obtain the generalized conjugate harmonic systems for the whole Clifford algebra-valued Hardy spaces rather than only the vector-valued cases in the Stein-Weiss setting.

A Hopf lemma for holomorphic functions in Hardy spaces and applications to CR mappings

We present results on a local version of Hopf’s Lemma and its application to the unique continuation problem for CR mappings between hypersurfaces.

Adaptive Fourier decomposition in Hp

In this paper, we study decomposition of functions in Hardy spaces . First, we will give a direct application of adaptive Fourier decomposition (AFD) of to functions in . Then, we study adaptive decomposition by the system (1)where Aa,p is the normalization factor making ea(z) to be of unit p-norm. Under the proposed decomposition procedure, we show that every can be effectively expressed by a linear combination of . We give a maximal selection principle of at the nth step and prove the convergence.

Adaptive Fourier decomposition in

In this paper, we study decomposition of functions in Hardy spaces . First, we will give a direct application of adaptive Fourier decomposition (AFD) of to functions in . Then, we study adaptive decomposition by the system urn:x-wiley:mma:media:mma5494:mma5494-math-0007 where Aa,p is the normalization factor making ea(z) to be of unit p‐norm. Under the proposed decomposition procedure, we show that every can be effectively expressed by a linear combination of . We give a maximal selection principle of at the nth step and prove the convergence.

Hardy space decompositions of Lp(ℝn) for 0 < p < 1 with rational approximation

ABSTRACTThis paper aims to obtain decompositions of higher dimensional functions into sums of non-tangential boundary limits of the corresponding Hardy space functions on tubes for the index range . In the one-dimensional case, Deng and Qian recently obtained such a Hardy space decomposition result: for any function , there exist functions and such that , where and are, respectively, the non-tangential boundary limits of some Hardy space functions in the upper-half and lower-half planes. In the present paper, we generalize the one-dimensional Hardy space decomposition result to the higher dimensions and discuss the uniqueness issue of such decomposition.

The Best Linear Approximation Methods of Analytic Functions in the disk and Exact Values of Widths of Some Classes Functions in Hardy Spaces

The Best Linear Approximation Methods of Analytic Functions in the disk and Exact Values of Widths of Some Classes Functions in Hardy Spaces

Hardy's Inequality for Functions of Several Complex Variables

We obtain a generalization of Hardy’s inequality for functions in the Hardy space H1 (Bd), where Bd is the unit ball {z = (z1, …, zd) ∈ In particular, we construct a function φ on the set of d –dimensional multi-indices {n = (n1, …, nd) | ni ∈   {0}} and prove that if f(z) = Σ anzn is a function in H1 (Bd), then ≤ Moreover, our proof shows that this inequality is also valid for functions in Hardy space on the polydisk H1 (Bd).

Which linear operators preserve outer functions?

The aim of this study is to determine the systems that preserve the minimum phase property in signal processing. The minimum phase signals are closely related to outer functions. A basic mathematical question that arises in geophysical imaging is to characterize the linear operators preserving the set of outer functions in Hardy spaces. It is shown that a bounded linear operator on the Hardy space Hp, 1<p<∞, preserving the set of outer functions is necessarily a weighted composition operator. Moreover, an operator preserving the set of shifted outer functions is necessarily a weighted composition operator as well. These results complement work by Gibson and Lamoureux.

Hardy-Type Spaces Arising from a Vector-Valued Cauchy Kernel

An integral formula of Cauchy type was recently developed that reproduces any continuous \(f:\overline{{\mathbb {B}}} \rightarrow {\mathbb {C}}^n\) that is holomorphic in the open unit ball \({\mathbb {B}}\) of \({\mathbb {C}}^n\) using a fixed vector-valued kernel and the scalar expression \(\langle f(u),u \rangle \), where \(u\in \partial {\mathbb {B}}\) and \(\langle \cdot ,\cdot \rangle \) is the Hermitian inner product in \({\mathbb {C}}^n\), which is key to defining the numerical range of f. We consider Hardy-type spaces associated with this vector-valued kernel. In particular, we introduce spaces of vector-valued holomorphic mappings properly containing the vector-valued Hardy spaces that are reproduced through the process described above and isomorphic spaces of scalar-valued non-holomorphic functions that satisfy many of the familiar properties of Hardy space functions. In the spirit of providing a straightforward introduction to these spaces, proof techniques have been kept as elementary as possible. In particular, the theory of maximal functions and singular integrals is avoided.

Uniqueness Properties of Hardy Space Functions

We study the boundary uniqueness properties of Hardy space functions in several complex variables. Along the way, we develop properties of the Lumer Hardy space.

Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables

Let $\mathcal{H}_\infty$ be the set of all ordinary Dirichlet series $D=\sum_n a_n n^{-s}$ representing bounded holomorphic functions on the right half plane. A multiplicative sequence $(b_n)$ of complex numbers is said to be an $\ell_1$-multiplier for $\mathcal{H}_\infty$ whenever $\sum_n |a_n b_n| < \infty$ for every $D \in \mathcal{H}_\infty$. We study the problem of describing such sequences $(b_n)$ in terms of the asymptotic decay of the subsequence $(b_{p_j})$, where $p_j$ denotes the $j$th prime number. Given a multiplicative sequence $b=(b_n)$ we prove (among other results): $b$ is an $\ell_1$-multiplier for $\mathcal{H}_\infty$ provided $|b_{p_j}| < 1$ for all $j$ and $\overline{\lim}_n \frac{1}{\log n} \sum_{j=1}^n b_{p_j}^{*2} < 1$, and conversely, if $b$ is an $\ell_1$-multiplier for $\mathcal{H}_\infty$, then $|b_{p_j}| < 1$ for all $j$ and $\overline{\lim}_n \frac{1}{\log n} \sum_{j=1}^n b_{p_j}^{*2} \leq 1$ (here $b^*$ stands for the decreasing rearrangement of $b$). Following an ingenious idea of Harald Bohr it turns out that this problem is intimately related with the question of characterizing those sequences $z$ in the infinite dimensional polydisk $\mathbb{D}^\infty$ (the open unit ball of $\ell_\infty$) for which every bounded and holomorphic function $f$ on $\mathbb{D}^\infty$ has an absolutely convergent monomial series expansion $\sum_{\alpha} \frac{\partial_\alpha f(0)}{\alpha!} z^\alpha$. Moreover, we study analogous problems in Hardy spaces of Dirichlet series and Hardy spaces of functions on the infinite dimensional polytorus $\mathbb{T}^\infty$.

A Frame Theory of Hardy Spaces with the Quaternionic and the Clifford Algebra Settings

The purpose of this article is twofold. The first is to construct frames of the $$L^2(\mathbb R^n)$$ by using dilation and modulation starting from a single function of a certain type. The second is to construct frames of the $$H^2(\mathbb R^n_{1,+})$$ by using a Cauchy type integral. The work is motivated by the recent development of sparse representation of Hardy space functions, and, especially, by adaptive Fourier decomposition in relation to rational orthogonal systems. We work in two contexts. One is the quaternionic space and the other is the Euclidean space in the Clifford algebra setting. We also investigate what type of functions can give rise to frames of $$L^2(\mathbb R^n)$$ by dilation and modulation.

Potential theoretic approach to design of accurate formulas for function approximation in symmetric weighted Hardy spaces

We propose a method for designing accurate interpolation formulas on the real axis for the purpose of function approximation in weighted Hardy spaces. In particular, we consider the Hardy space of functions that are analytic in a strip region around the real axis, being characterized by a weight function $w$ that determines the decay rate of its elements in the neighborhood of infinity. Such a space is considered as a set of functions that are transformed by variable transformations that realize a certain decay rate at infinity. Popular examples of such transformations are given by the single exponential (SE) and double exponential (DE) transformations for the SE-Sinc and DE-Sinc formulas, which are very accurate owing to the accuracy of sinc interpolation in the weighted Hardy spaces with single and double exponential weights $w$, respectively. However, it is not guaranteed that the sinc formulas are optimal in weighted Hardy spaces, although Sugihara has demonstrated that they are near optimal. An explicit form for an optimal approximation formula has only been given in weighted Hardy spaces with SE weights of a certain type. In general cases, explicit forms for optimal formulas have not been provided so far. We adopt a potential theoretic approach to obtain almost optimal formulas in weighted Hardy spaces in the case of general weight functions $w$. We formulate the problem of designing an optimal formula in each space as an optimization problem written in terms of a Green potential with an external field. By solving the optimization problem numerically, we obtain an almost optimal formula in each space. Furthermore, some numerical results demonstrate the validity of this method. In particular, for the case of a DE weight, the formula designed by our method outperforms the DE-Sinc formula.