Holder Class Research Articles

Homotopy Perturbation Method for Semi-Bounded Solution of the System of Cauchy-Type Singular Integral Equations of the First Kind

In this paper, we introduce the standard Homotopy Perturbation Method (HPM) for obtaining semi-bounded solutions of the first kind of system of Cauchy-type singular integral equations (CSIEs) with constant coefficients. We use the Gauss elimination technique to reduce the system of CSIEs to a diagonal triangle system of algebraic equations. We then apply the HPM to solve the resulting equations. By applying the theory of semi-bounded solutions of CSIEs, we can determine the inverse operators for the first kind of CSIEs. We demonstrate that the proposed method is exact for the system of characteristic SIEs, regardless of the choice of initial guesses (in the Holder class of functions). To illustrate the validity and accuracy of our proposed method, we supply and analyse three examples. We compare the results obtained using our method with those obtained using the Chebyshev collocation and Galerkin methods. Our method includes the ability to solve the complex-valued system of CSIEs. Based on the numerical results, we conclude that the HPM is more dominant than the other methods.

Neural Network Approximation for Pessimistic Offline Reinforcement Learning

Deep reinforcement learning (RL) has shown remarkable success in specific offline decision-making scenarios, yet its theoretical guarantees are still under development. Existing works on offline RL theory primarily emphasize a few trivial settings, such as linear MDP or general function approximation with strong assumptions and independent data, which lack guidance for practical use. The coupling of deep learning and Bellman residuals makes this problem challenging, in addition to the difficulty of data dependence. In this paper, we establish a non-asymptotic estimation error of pessimistic offline RL using general neural network approximation with C-mixing data regarding the structure of networks, the dimension of datasets, and the concentrability of data coverage, under mild assumptions. Our result shows that the estimation error consists of two parts: the first converges to zero at a desired rate on the sample size with partially controllable concentrability, and the second becomes negligible if the residual constraint is tight. This result demonstrates the explicit efficiency of deep adversarial offline RL frameworks. We utilize the empirical process tool for C-mixing sequences and the neural network approximation theory for the Holder class to achieve this. We also develop methods to bound the Bellman estimation error caused by function approximation with empirical Bellman constraint perturbations. Additionally, we present a result that lessens the curse of dimensionality using data with low intrinsic dimensionality and function classes with low complexity. Our estimation provides valuable insights into the development of deep offline RL and guidance for algorithm model design.

A Hybrid Method for All Types of Solutions of the System of Cauchy-Type Singular Integral Equations of the First Kind

In this note, the hybrid method (combination of the homotopy perturbation method (HPM) and the Gauss elimination method (GEM)) is developed as a semi-analytical solution for the first kind system of Cauchy-type singular integral equations (CSIEs) with constant coefficients. Before applying the HPM, we have to first reduce the system of CSIEs into a triangle system of algebraic equations using GEM, which is then carried out using the HPM. Using the theory of the bounded, unbounded and semi-bounded solutions of CSIEs, we are able to find inverse operators for the system of CSIEs of the first kind. A stability analysis and convergent of the proposed method has been conducted in the weighted Lp space. Moreover, the proposed method is proven to be exact in the Holder class of functions for the system of characteristic SIEs for any type of initial guess. For each of the four cases, several examples are provided and examined to demonstrate the proposed method’s validity and accuracy. Obtained results are compared with the Chebyshev collocation method and modified HPM (MHPM). Example 3 reveals that the error term of the MHPM is slightly superior to that of the HPM. One of the features of the proposed method is that it can be solved as a complex-valued system of CSIEs. Numerical results revealed that the hybrid method dominates others.

Some applied aspects of the Dirac delta function

The study is devoted to some applied aspects of the Dirac delta function. On the basis of this function, an integral representation was found for the deviation of the functions of the Holder class ${H}^{\alpha }$ ($0<\alpha <1$) from their Poisson integrals in the upper half-plane. In the current research, exact equalities of the upper bounds for the deviations of the functions of the Holder class ${H}^{\alpha }$ from the Poisson operators in the upper half-plane were found by applying the known properties of the Dirac delta function.

Boundedness on generalized Morrey spaces for the Schrodinger operator with potential in a reverse Holder class

In this article, we prove boundedness for the Hessian of a Schrodinger operator with weak regularity on the coefficients, and potentials satisfying the reverse H\"older condition. This is done in in generalized Morrey spaces, and in vanishing generalized Morrey spaces. On the Schrodinger operator \(L=-a_{ij}(x)D_{ij}+V(x)\) it is assumed that \(a_{ij}\in \rm{BMO}_{\theta}(\rho)\) (a generalized Morrey space) and that \(V(x)\in B^*_{n/2}\) (a reverse Holder class).
 For more information see https://ejde.math.txstate.edu/Volumes/2023/67/abstr.html

Estimation of the invariant density for discretely observed diffusion processes: impact of the sampling and of the asynchronicity

We aim at estimating in a non-parametric way the density π of the stationary distribution of a d-dimensional stochastic differential equation , for , from the discrete observations of a finite sample ,…, with . We propose a kernel density estimator and we study its convergence rates for the pointwise estimation of the invariant density under anisotropic Hölder smoothness constraints. First of all, we find some conditions on the discretization step that ensures it is possible to recover the same rates as if the continuous trajectory of the process was available. As proven in the recent work [Amorino C, Gloter A. Minimax rate of estimation for invariant densities associated to continuous stochastic differential equations over anisotropic Holder classes; 2021. arXiv preprint arXiv:2110.02774], such rates are optimal and new in the context of density estimator. Then we deal with the case where such a condition on the discretization step is not satisfied, which we refer to as the intermediate regime. In this new regime we identify the convergence rate for the estimation of the invariant density over anisotropic Hölder classes, which is the same convergence rate as for the estimation of a probability density belonging to an anisotropic Hölder class, associated to n iid random variables . After that we focus on the asynchronous case, in which each component can be observed at different time points. Even if the asynchronicity of the observations complexifies the computation of the variance of the estimator, we are able to find conditions ensuring that this variance is comparable to the one of the continuous case. We also exhibit that the non-synchronicity of the data introduces additional bias terms in the study of the estimator.

Processing methods of donor human milk evaluated by a blood plasma clotting assay

Donor human milk is the first alternative for preterm infants when mother's own milk is not available. Most available human milk banking guidelines recommend classical holder pasteurization to ensure safety by eliminating potential infectious microorganisms. Processing by heat treatment, however, negatively affects functionality and availability of bioactive components naturally present in human milk. Here we compared the effect of five different processing methods on the ability of human milk to induce blood plasma clotting, which was recently described as a bioactive function present in human milk. From thirty lactating women, milk samples were collected, and all milk samples were subjected to holder pasteurization (30 min at 62.5 °C), high-temperature-short-time pasteurization (15 s at 72 °C), high-pressure processing (5 min at 500 MPa), ultraviolet-C irradiation (4863 J/L), or thermo-ultrasonication (6 min at 60 W, at 40 °C). All methods significantly reduced the ability of milk to trigger blood plasma clotting compared to untreated milk, but ultraviolet-C irradiation and high-pressure processing were best at preserving this activity. Taken together, measuring the ability of milk to induce blood plasma clotting may offer a new tool to monitor the effect of human milk processing.

Fractional Integral Inequalities for Some Convex Functions

In this paper, we obtained several new integral inequalities using fractional Riemann-Liouville integrals for convex s-Godunova-Levin functions in the second sense and for quasi-convex functions. The results were gained by applying the double Hermite-Hadamard inequality, the classical Holder inequalities, the power mean, and weighted Holder inequalities. In particular, the application of the results for several special computing facilities is given. Some applications to special means for arbitrary real numbers: arithmetic mean, logarithmic mean, and generalized log-mean, are provided.

Constructive Description of Hölder Classes on Some Multidimensional Compact Sets

In this paper, a constructive description of Holder classes of functions on certain compact sets in $${{\mathbb{R}}^{m}}$$ (m $$ \geqslant $$ 3) is given in terms of the rate of approximation by harmonic functions in shrinking neighborhoods of these compact sets. The compact sets under consideration are a higher-dimensional generalization of compact subsets of curves in $${{\mathbb{R}}^{3}}$$ whose arc is commensurable with chord. The neighborhood size is directly connected with the rate of approximation: it shrinks when the approximation is getting more accurate. In addition to being harmonic in the neighborhood of the compact, the approximating functions are subject to the condition that looks similar to the Holder condition, but is weaker. Namely, the difference in values at two points is estimated in terms of the size of the neighborhood, if the distance between these points is commensurate with the size of the neighborhood (and thus estimated in terms of the distance between the points).

Solution of the Dirichlet Problem for the Inhomogeneous Lamé System with Lower Order Coefficients

We consider the Dirichlet problem for the inhomogeneous Lame system in the plane with constant leading coefficients in a (finite or infinite) domain, bounded by a Lyapunov contour. For domains of finite diameter we use the weighted Holder class of functions with power behavior at infinity. We propose an equivalent reduction of the problem to a system of Fredholm integral equations in the domain and on the boundary contour.

Correction note: “Optimal two-stage procedures for estimating location and size of the maximum of a multivariate regression function” Ann. Statist. 40 (2012) 2850–2876

In the paper of Belitser, Ghosal and van Zanten (Ann. Statist. 40 (2012) 2850-2876), on page 2851 it is stated that “the minimax rate for estimating the maximum value of the function ranging over an α-smooth nonparametric class (e.g., isotropic Holder class defined below) is n−α/(2α+d)” (the rate n−α/(2α+d) is also mentioned in abstract, on pp. 2553 and 2859). This is not correct: instead of “n−α/(2α+d)” one should read “n−α/(2α+d) up to a log factor.” Recall the model Yk = f(xk)+ ξk, xk ∈ D ⊂ Rd, k = 1,...,n, the assumptions and notation (like, ≲, etc.) from Belitser et al. (2012). Consider this model now under a fixed equidistant design xk ∈ D ⊂ Rd, k = 1,...,n. Define R(Hd(α,D)) = inf M f∈Hd(α,D) sup Ef|M − Mf|, Mf = sup f(x), x∈D the infimum is taken over all estimators. The results of [1-3] suggest that (1) R(Hd(α,D)) (n/logn)−α/(2α+d). We call (1) conjecture for now, because, despite our extensive search in the literature, we were unable to find an exact reference that establishes (or implies) (1). The results [1] and [3] are obtained only for the one dimensional case in the white noise model, so formally they do not provide a conclusive proof of (1). Below we provide a proof for the upper bound. The lower bound should proceed in the same way as for the problem of estimating the function in the sup-norm. In essence, the difficulty of the problem of estimating the maximal values of a function is the same as that of the problem of estimating the function in the sup-norm. Here, we provide a short argument for the logarithmic sandwich bound n−α/(2α+d) ≲ R(Hd(α,D)) ≲ (n/logn)−α/(2α+d) (2). The first inequality of (2) can be argued by comparing the problem of estimating the maximal values of a function to the problem of estimating a function value at a fixed point x0 ∈ D, the so-called pointwise estimation problem. The pointwise minimax rate is known to be Rpw(Hd(α,D)) = inf sup Ef ||f( x0)− f(x0)|| n−α/(2α+d). f( x0)f∈Hd(α,D)learly, the former problem is not easier than the latter, which implies the first relation in (2): n−α/(2α+d) ≲ Rpw(Hd(α,D))≲ R(Hd(α,D)).

On the existence of global solution of the system of equations of liquid movement in porous medium

The initial-boundary value problem for the system of one-dimensional isothermal motion of viscous liquid in deformable viscous porous medium is considered. Local theorem of existence and uniqueness of problem is proved in case of compressible liquid. In case of incompressible liquid the theorem of global solvability in time is proved in Holder classes. A feature of the model of fluid filtration in a porous medium considered in this paper is the inclusion of the mobility of the solid skeleton and its poroelastiс properties. The transition from Euler variables to Lagrangian variables is used in the proof of the theorems.

Rate of estimation for the stationary distribution of jump-processes over anisotropic Holder classes

We study the problem of the non-parametric estimation for the density of the stationary distribution of the multivariate stochastic differential equation with jumps (Xt) , when the dimension d is bigger than 3. From the continuous observation of the sampling path on [0, T ] we show that, under anisotropic Holder smoothness constraints, kernel based estimators can achieve fast convergence rates. In particular , they are as fast as the ones found by Dalalyan and Reiss [9] for the estimation of the invariant density in the case without jumps under isotropic Holder smoothness constraints. Moreover, they are faster than the ones found by Strauch [29] for the invariant density estimation of continuous stochastic differential equations, under anisotropic Holder smoothness constraints. Furthermore, we obtain a minimax lower bound on the L2-risk for pointwise estimation, with the same rate up to a log(T) term. It implies that, on a class of diffusions whose invariant density belongs to the anisotropic Holder class we are considering, it is impossible to find an estimator with a rate of estimation faster than the one we propose.

Optimal inference with a multidimensional multiscale statistic

We observe a stochastic process $Y$ on $[0,1]^d$ ($d\geq 1$) satisfying $dY(t)=n^{1/2}f(t)dt$ + $dW(t)$, $t \in [0,1]^d$, where $n \geq 1$ is a given scale parameter (`sample size'), $W$ is the standard Brownian sheet on $[0,1]^d$ and $f \in L_1([0,1]^d)$ is the unknown function of interest. We propose a multivariate multiscale statistic in this setting and prove its almost sure finiteness; this extends the work of Dumbgen and Spokoiny (2001) who proposed the analogous statistic for $d=1$. We use the proposed multiscale statistic to construct optimal tests for testing $f=0$ versus (i) appropriate Holder classes of functions, and (ii) alternatives of the form $f=\mu_n \mathbb{I}_{B_n}$, where $B_n$ is an axis-aligned hyperrectangle in $[0,1]^d$ and $\mu_n \in \mathbb{R}$; $\mu_n$ and $B_n$ unknown. In the process we generalize Theorem 6.1 of Dumbgen and Spokoiny (2001) about stochastic processes with sub-Gaussian increments on a pseudometric space, which is of independent interest.

Numerical Solution of Nonlinear Fredholm and Volterra Integrals by Newton–Kantorovich and Haar Wavelets Methods

The current study proposes a numerical method which solves nonlinear Fredholm and Volterra integral of the second kind using a combination of a Newton–Kantorovich and Haar wavelet. Error analysis for the Holder classes was established to ensure convergence of the Haar wavelets. Numerical examples will illustrate the accuracy and simplicity of Newton–Kantorovich and Haar wavelets. Numerical results of the current method were then compared with previous well-established methods.

Schwarz Problem in Ellipse for Nondiagonalizable Matrices

We study the Schwarz problem for J-analytic vector-valued functions in an ellipse with a square matrix J admitting a nondiagonal Jordan form. We obtain conditions on the ellipse and matrix J necessary and sufficient for the existence and uniqueness of a solution to the Schwarz problem with an arbitrary boundary function of Holder class. Under certain conditions on the matrix J, we show that the homogeneous Schwarz problem in an ellipse has a solution in the form of a vector polynomial of an arbitrary degree.

Lp estimates for Baouendi–Grushin operators

We prove L^p estimates for the Baouendi-Grushin operator L=Delta_x+|x|^\alpha Delta_y in L^p(R^N+M), 1 < p < 1, where x belongs to R^N; y belongs to R^M. When p = 2 more general weights belonging to Reverse Holder classes are allowed.

Parabolic Hermite Lipschitz Spaces: Regularity of Fractional Operators

We introduce a pointwise definition of Lipschitz (also called Holder) spaces adapted to the parabolic Hermite operator $$\mathbb {H}= \partial _t- \Delta _x+|x|^2$$ on $$\mathbb {{R}}^{n+1}$$ . Also for every $$\alpha >0$$ , we define the following spaces by means of the Poisson semigroup of $$\mathbb {H}$$ , $$\mathcal {P}_y^{\mathbb {H}}=e^{-y\sqrt{\mathbb {H}}}$$ : $$\begin{aligned} \Lambda _\alpha ^{\mathcal {P}^\mathbb {H}}= & {} \left\{ f: \;f\in L^\infty (\mathbb {R}^{n+1})\, \mathrm{and} \, \left\| \partial _y^k e^{-y\sqrt{\mathbb {H}}} f \right\| _{L^\infty (\mathbb {R}^{n+1})}\right. \\&\left. \le C_k y^{-k+\alpha },\, \mathrm {for}\, k=[\alpha ]+1,\;y>0 \right\} , \end{aligned}$$ with the obvious norm. We prove that both spaces do coincide and their norms are equivalent. For the harmonic oscillator, $$\mathcal {{H}}=-\Delta _x+|x|^2$$ , Stinga and Torrea introduced in 2011 adapted Holder classes. Parallel to the parabolic case, we characterize these pointwise Holder spaces via the $$L^\infty $$ norm of the derivatives of the Poisson and heat semigroups, $$e^{-y\sqrt{\mathcal {{H}}}}$$ and $$e^{-\tau \mathcal {{H}}}$$ , respectively. As important applications of these semigroups characterizations, we get regularity results regarding the boundedness in these adapted Lipschitz spaces of operators related to $$\mathbb {H}$$ and $$\mathcal {{H}}$$ as fractional (positive and negative) powers, Bessel potentials, Hermite Riesz transforms, and Laplace transform multipliers, in a more direct way. The proofs use in a fundamental way the semigroup definition of the operators considered along the paper. The non-convolution structure of the operators produces an extra difficulty on the arguments.

Bayesian high-dimensional semi-parametric inference beyond sub-Gaussian errors

We consider a sparse linear regression model with unknown symmetric error under the high-dimensional setting. The true error distribution is assumed to belong to the locally $$\beta $$ -Holder class with an exponentially decreasing tail, which does not need to be sub-Gaussian. We obtain posterior convergence rates of the regression coefficient and the error density, which are nearly optimal and adaptive to the unknown sparsity level. Furthermore, we derive the semi-parametric Bernstein-von Mises (BvM) theorem to characterize asymptotic shape of the marginal posterior for regression coefficients. Under the sub-Gaussianity assumption on the true score function, strong model selection consistency for regression coefficients are also obtained, which eventually asserts the frequentist’s validity of credible sets.

Uniqueness of a Solution in a Hölder Class to the First Initial Boundary Value Problem for a Parabolic System in a Bounded Nonsmooth Domain in the Plane

We consider the first initial boundary value problem for a Petrovsky second order parabolic system with variable coefficients in a bounded plane domain with nonsmooth lateral boundary. We prove the uniqueness of a solution in a Holder class.