Hardy Functions Research Articles

Fundamental sequences and fast-growing hierarchies for the Bachmann-Howard ordinal

Hardy functions are defined by transfinite recursion and provide upper bounds for the growth rate of the provably total computable functions in various formal theories, making them an essential ingredient in many proofs of independence. Their definition is contingent on a choice of fundamental sequences, which approximate limits in a ‘canonical’ way. In order to ensure that these functions behave as expected, including the aforementioned unprovability results, these fundamental sequences must enjoy certain regularity properties.In this article, we prove that Buchholz's system of fundamental sequences for the ϑ function enjoys such conditions, including the Bachmann property. We partially extend these results to variants of the ϑ function, including a version without addition for countable ordinals. We conclude that the Hardy functions based on these notation systems enjoy natural monotonicity properties and majorize all functions defined by primitive recursion along ϑ(εΩ+1).

Multiplication between elements in martingale Hardy spaces and their dual spaces

In the first part of the paper, we establish continuous bilinear decompositions that arise in the study of products between elements in martingale Hardy spaces Hp(0<p⩽1) and functions in their dual spaces. Our decompositions are based on martingale paraproducts. The second part of the paper concerns applications of the method developed for the classical martingales in the first part. In particular, we build a connection between Hardy spaces on spaces of homogenous type equipped with a doubling measure and Hardy spaces with respect to the corresponding dyadic martingales. Using the method introduced in the first part, we obtain analogous results for dyadic martingales on spaces of homogenous type, which, thanks to the aforementioned connection, yield conclusions for products between elements in Hardy spaces and those in their duals on spaces of homogenous type. The key property of martingale Hardy spaces in the study is that they admit the atomic decomposition, for which we provide an interpretation via duality.

Hardy function modeling elevation anomaly based Tikhonov regularization

Hardy function modeling elevation anomaly based Tikhonov regularization

Pointwise convergence of multiple ergodic averages – PCMEA

Pointwise convergence of multiple ergodic averages – PCMEA Given an arbitrary measure preserving system we show that the multilinear ergodic averages sampled along an arbitrary number of sequences coming from a Hardy field converge pointwise almost everywhere. We aim to prove this for as wide a class of Hardy field functions as possible. To do so, we establish a long variational inequality along lacunary sequences which implies a maximal inequality, norm convergence, and pointwise convergence. By a transference argument it suffices prove this long variational inequality in the case that the measure preserving system is the integers. This reduction allows us to use tools from discrete harmonic analysis, additive combinatorics, and analytic number theory. We then give applications in areas such as upcrossings, equidistribution, and combinatorics.

A note on the zeros of the derivatives of Hardy's function Z(t)$Z(t)$

AbstractUsing the twisted fourth moment of the Riemann zeta‐function, we study large gaps between consecutive zeros of the derivatives of Hardy's function , improving upon previous results of Conrey and Ghosh (J. Lond. Math. Soc. 32 (1985) 193–202), and of the second named author (Acta Arith. 111 (2004) 125–140). We also exhibit small distances between the zeros of and the zeros of for every , in support of our numerical observation that the zeros of and , when k and ℓ have the same parity, seem to come in pairs that are very close to each other. The latter result is obtained using the mollified discrete second moment of the Riemann zeta‐function.

On the derivatives of Hardy's function Z(t)

AbstractLet be the th derivative of Hardy's ‐function. The numerics seem to suggest that if and have the same parity, then the zeros of and come in pairs that are very close to each other. That is to say that has constant sign for the majority, if not almost all, of values . In this paper, we show that this is true a positive proportion of times. We also study the sign of the product of four derivatives of Hardy's function, .

The affine ensemble: determinantal point processes associated with the $ax + b$ group

We introduce the affine ensemble, a class of determinantal point processes (DPP) in the half-plane $\mathbb{C}^{+}$ associated with the $ax + b$ (affine) group, depending on an admissible Hardy function $\psi$. We obtain the asymptotic behavior of the variance, the exact value of the asymptotic constant, and non-asymptotic upper and lower bounds for the variance on a compact set $\Omega \subset \mathbb{C}^{+}$. As a special case one recovers the DPP related to the weighted Bergman kernel. When $\psi$ is chosen within a finite family whose Fourier transform are Laguerre functions, we obtain the DPP associated to hyperbolic Landau levels, the eigenspaces of the finite spectrum of the Maass Laplacian with a magnetic field.

Representations of generalized Hardy functions in Beurling’s tempered distributions

Let B be a proper open subset in $${{\mathbb {R}}}^N$$ and C be a regular cone in $${{\mathbb {R}}}^N$$ . On our previous paper of Acta Scientiarum Mathematicarum 85, 595–611 (2019), we have defined the space of generalized Hardy functions, $$G_{\omega ^*,A}^p(T^B)$$ , $$1< p \le 2,$$ and $$A \ge 0$$ , and have shown that the functions in $$G_{\omega ^*,A}^p(T^B)$$ have distributional boundary values in the weak topology of Beurling tempered distributions, $${\mathcal {S}}_{(\omega )}^\prime $$ . In this paper we show that if the distributional boundary values are convolutors in Beurling ultradistributions of $$L_2$$ -growth, then the functions in $$G_{\omega ^*,0}^p(T^C)$$ , $$1< p \le 2,$$ can be represented as Cauchy and Poisson integral of the boundary values in $${\mathcal {S}}_{(\omega )}^\prime $$ .

Two Coefficient Conjectures for Nonvanishing Hardy Functions, I

There are two eminent still open conjectures for nonvanishing holomorphic Hardy functions f(z) on the unit disk. The Hummel–Scheinberg–Zalcman conjecture posed in 1977 extends Krzyz’s conjecture of 1968 to Hp spaces with finite p > 1 and states that Taylor’s coefficients of nonvanishing holomorphic functions f ∈ Hp with norm ‖f‖p ≤ 1 are sharply estimated by |cn| ≤ (2/e)1 − 1/p, with appropriate extremal functions. Both conjectures have been investigated by many authors; however still remain open. The desired Krzyz’s estimate |cn| ≤ 2/e for f ∈ H∞ with ‖f‖∞ ≤ 1 was established only for the initial coefficients cn with n ≤ 5 The only known results for the Hummel–Scheinberg–Zalcman conjecture are that it is true for n = 1 and n = 2 as well as some results for special subclasses of Hp. We prove here that the Hummel–Scheinberg–Zalcman conjecture is true for all spaces H2m with m ∈ N. In the limit as m → ∞, this also provides the proof of Krzyz’s conjecture. Our approach involves deep results from Teichmüller space theory, especially the Bers isomorphism theorem for Teichmüller spaces of punctured Riemann surfaces, and special quasiconformal deformations of H2m functions.

Arithmetical and Hyperarithmetical Worm Battles

Abstract Japaridze’s provability logic ${\operatorname {GLP}}$ has one modality $[n]$ for each natural number and has been used by Beklemishev for a proof theoretic analysis of Peano arithmetic (${\operatorname {PA}}$) and related theories. Among other benefits, this analysis yields the so-called Every Worm Dies (${\operatorname {EWD}}$) principle, a natural combinatorial statement independent of ${\operatorname {PA}}$. Recently, Beklemishev and Pakhomov have studied notions of provability corresponding to transfinite modalities in ${\operatorname {GLP}}$. We show that indeed the natural transfinite extension of ${\operatorname {GLP}}$ is sound for this interpretation and yields independent combinatorial principles for the second-order theory ${\operatorname {ACA}}$ of arithmetical comprehension with full induction. We also provide restricted versions of ${\operatorname {EWD}}$ related to the fragments ${\operatorname {I\varSigma }}_n$ of PA. In order to prove the latter, we show that standard Hardy functions majorize their variants based on tree ordinals.

Generalized Vector-Valued Hardy Functions

We consider analytic functions in tubes Rn+iB⊂Cn with values in Banach space or Hilbert space. The base of the tube B will be a proper open connected subset of Rn, an open connected cone in Rn, an open convex cone in Rn, and a regular cone in Rn, with this latter cone being an open convex cone which does not contain any entire straight lines. The analytic functions satisfy several different growth conditions in Lp norm, and all of the resulting spaces of analytic functions generalize the vector valued Hardy space Hp in Cn. The analytic functions are represented as the Fourier–Laplace transform of certain vector valued Lp functions which are characterized in the analysis. We give a characterization of the spaces of analytic functions in which the spaces are in fact subsets of the Hardy functions Hp. We obtain boundary value results on the distinguished boundary Rn+i{0¯} and on the topological boundary Rn+i∂B of the tube for the analytic functions in the Lp and vector valued tempered distribution topologies. Suggestions for associated future research are given.

Two coefficient conjectures for nonvanishing Hardy functions, I

Two coefficient conjectures for nonvanishing Hardy functions, I

On a distinguished family of random variables and Painlevé equations

A family of random variables $\mathbf{X}(s)$, depending on a real parameter $s>-\frac{1}{2}$, appears in the asymptotics of the joint moments of characteristic polynomials of random unitary matrices and their derivatives, in the ergodic decomposition of the Hua-Pickrell measures and conjecturally in the asymptotics of the joint moments of Hardy's function and its derivative. Our first main result establishes a connection between the characteristic function of $\mathbf{X}(s)$ and the $\sigma$-Painlev\'e III' equation in the full range of parameter values $s>-\frac{1}{2}$. Our second main result gives the first explicit expression for the density and all the complex moments of the absolute value of $\mathbf{X}(s)$ for integer values of $s$. Finally, we establish an analogous connection to another special case of the $\sigma$-Painlev\'e III' equation for the Laplace transform of the sum of the inverse points of the Bessel point process.

Sparse Hardy function model of regional velocity field from GNSS data

Classical Hardy function interpolation method is often necessary in establishment of regional velocity field. In this work, a regularization method, namely the L1-norm regularization, also called the least absolute shrinkage selection operator (Lasso), is employed to improve the traditional method. With the new method, a sparse model can be obtained with many zero elements in the parameter vector. Compared to L2-norm regularization, which is also called the Tikhonov regularization, the L1-norm regularization will select the best model automatically by making the coefficient of the unnecessary kernels zero, through solving a convex optimization problem. In this paper, the velocity field dataset derived from global navigation satellite system data is used to establish models in different directions. By comparing the interpolation accuracy of velocity at the same unknown points of Hardy function improved by Lasso and Tikhonov regularizations respectively, the feasibility of the former is verified. The results show that L1-norm regularization method has a slightly worse interpolation accuracy than Tikhonov regularization method in North and Up directions, but in East direction, the interpolation effect is much better and all directions can get great sparse performance. More specifically, the prediction accuracy of the proposed method in the East direction is improved by 17.3%, but in North and Up directions is decreased by 5.4% and 6.3% respectively in comparison with that of Tikhonov regularization method, though the sparse rates of them are over 60%. In addition, besides selecting the root mean square error as the evaluation standard for model selection, we can also select the sparse rate as the evaluation criteria to make the model much sparser if necessary. A sparse model would be beneficial in terms of better interpretability and improved variable assigning efficiency. To summarize, L1-norm regularization can be viewed as a potential alternative in velocity field modelling.

Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg System

The asymptotic profile for diffusion wave terms of solutions to the compressible Navier–Stokes–Korteweg system is studied on R2. The diffusion wave with time-decay estimate was studied by Hoff and Zumbrun (1995, 1997), Kobayashi and Shibata (2002), and Kobayashi and Tsuda (2018) for compressible Navier–Stokes and compressible Navier–Stokes–Korteweg systems. Our main assertion in this paper is that, for some initial conditions given by the Hardy space, asymptotic behaviors in space–time L2 of the diffusion wave parts are essentially different between density and the potential flow part of the momentum. Even though measuring by L2 on space, decay of the potential flow part is slower than that of the Stokes flow part of the momentum. The proof is based on a modified version of Morawetz’s energy estimate, and the Fefferman–Stein inequality on the duality between the Hardy space and functions of bounded mean oscillation.

Decomposition of $L^{2}$-Vector Fields on Lipschitz Surfaces: Characterization via Null-Spaces of the Scalar Potential

For $\partial\Omega$ the boundary of a bounded and connected strongly Lipschitz domain in $\mathbb{R}^{d}$ with $d\geq3$, we prove that any field $f\in L^{2}(\partial\Omega ; \mathbb{R}^{d})$ decomposes, in a unique way, as the sum of three invisible vector fields---fields whose magnetic potential vanishes in one or both components of $\mathbb{R}^d\setminus\partial\Omega$. Moreover, this decomposition is orthogonal if and only if $\partial\Omega$ is a sphere. We also show that any $f$ in $L^{2}(\partial\Omega ; \mathbb{R}^{d})$ is uniquely the sum of two invisible fields and a Hardy function, in which case the sum is orthogonal regardless of $\partial\Omega$; we express the corresponding orthogonal projections in terms of layer potentials. When $\partial\Omega$ is a sphere, both decompositions coincide and match what has been called the Hardy--Hodge decomposition in the literature.

On real zeros of the derivative of the Hardy function

On real zeros of the derivative of the Hardy function

Power bounded operators and the mean ergodic theorem for subsequences

Let T be a power bounded Hilbert space operator without unimodular eigenvalues. We show that the subsequential ergodic averages N−1∑n=1NTan converge in the strong operator topology for a wide range of sequences (an), including the integer part of most of subpolynomial Hardy functions. Moreover, we show that the weighted averages N−1∑n=1Ne2πig(n)Tan also converge for many reasonable functions g. In particular, we generalize the polynomial mean ergodic theorem for power bounded operators due to ter Elst and the second author [16] to real polynomials and polynomial weights.

Multiple ergodic averages for tempered functions

Following Frantzikinakis' approach on averages for Hardy field functions of different growth, we add to the topic by studying the corresponding averages for tempered functions, a class which also contains functions that oscillate and is in general more restrictive to deal with. Our main result is the existence and the explicit expression of the \begin{document}$ L^2 $\end{document} -norm limit of the aforementioned averages, which turns out, as in the Hardy field case, to be the expected one. The main ingredients are the use of, the now classical, PET induction (introduced by Bergelson), covering a more general case, namely a nice class of tempered functions (developed by Chu-Frantzikinakis-Host for polynomials and Frantzikinakis for Hardy field functions) and some equidistribution results on nilmanifolds (analogous to the ones of Frantzikinakis' for the Hardy field case).

An approximation theorem for Hardy functions on $\mathbb {C}$-linearly convex domains of infinite type in $\mathbb {C}^2$

Let $\Omega \subset \mathbb {C}^2$ be a smoothly bounded, $\mathbb {C}$-linearly convex domain. We prove that if $\Omega $ is of $F$-type at all boundary points (for some type function $F$), then for all $1\le p \lt \infty $, every Hardy function $f\in H^