Lder Classes Research Articles

Solution of Abel's integral equation by modified Taylor wavelet with error analysis

In this paper, a modified Taylor wavelet (MTW) is introduced. Three new estimators $ E_{2^{k-1},M}^{(1)}(f) $, $ E_{2^{k-1},M}^{(2)}(f) $ and $ E_{2^{k-1},M}^{(3)}(f) $ of functions of H$ \ddot{\text{o}} $lder's class $ H^{\alpha} [0,1) $ of order $ 0<\alpha \leq 1 $ have been established by MTW method. A MTW method is presented for the numerical solution of Abel's integral equations. MTW operational matrices are formulated. These matrices have been utilized for solving Abel's integral equations. Some examples are included to demonstrate the validity and efficiency of the proposed method.

Approximation of functions of H$$\ddot{o}$$lder class and solution of ODE and PDE by extended Haar wavelet operational matrix

Approximation of functions of H$$\ddot{o}$$lder class and solution of ODE and PDE by extended Haar wavelet operational matrix

Approximation of Function in generalized Hölder Class

IIn the present work, we study error estimation of a function g ∈ H(η) r (r ≥ 1) class using Matrix-Hausdorff (T ∆H) means of its Fourier series. Our Theorem 1 generalizes twelve previously known results. Thus, the results of [4-5, 11–16, 18, 26, 29-30] become the particular cases of our Theorem 1. Several useful results in the form of corollaries are also deduced from our Theorem 1.

Nonparametric estimation of low rank matrix valued function

Let $A:[0,1]\rightarrow\mathbb{H}_m$ (the space of Hermitian matrices) be a matrix valued function which is low rank with entries in H\"{o}lder class $\Sigma(\beta,L)$. The goal of this paper is to study statistical estimation of $A$ based on the regression model $\mathbb{E}(Y_j|\tau_j,X_j) = \langle A(\tau_j), X_j \rangle,$ where $\tau_j$ are i.i.d. uniformly distributed in $[0,1]$, $X_j$ are i.i.d. matrix completion sampling matrices, $Y_j$ are independent bounded responses. We propose an innovative nuclear norm penalized local polynomial estimator and establish an upper bound on its point-wise risk measured by Frobenius norm. Then we extend this estimator globally and prove an upper bound on its integrated risk measured by $L_2$-norm. We also propose another new estimator based on bias-reducing kernels to study the case when $A$ is not necessarily low rank and establish an upper bound on its risk measured by $L_{\infty}$-norm. We show that the obtained rates are all optimal up to some logarithmic factor in minimax sense. Finally, we propose an adaptive estimation procedure based on Lepskii's method and model selection with data splitting which is computationally efficient and can be easily implemented and parallelized.

Mutual estimates for the dyadic reverse Hölder and Muckenhoupt constants for the dyadically doubling weights

Muckenhoupt and Reverse H\"{o}lder classes of weights play an important role in harmonic analysis, PDE's and quasiconformal mappings. In 1974 Coifman and Fefferman showed that a weight belongs to a Muckenhoupt class $A_p$ for some $1<p<\infty$ if and only if it belongs to a Reverse H\"{o}lder class $RH_q$ for some $1<q<\infty$. In 2009 Vasyunin found the exact dependence between $p$, $q$ and the corresponding characteristic of the weight using the Bellman function method. The result of Coifman and Fefferman works for the dyadic classes of weights under an additional assumption that the weights are dyadically doubling. We extend the Vasyunin's result to the dyadic Reverse H\"{o}lder and Muckenhoupt classes and obtain the dependence between $p$, $q$, the doubling constant and the corresponding characteristic of the weight. More precisely, given a dyadically doubling weight in $RH_p^d$ on a given dyadic interval $I$, we find an upper estimate on the average of the function $w^{q}$ over the interval $I$. From the bound on this average we can conclude, for example, that $w$ belongs to the corresponding $A_{s_1}^d$ class or that $w^p$ is in $A_{s_2}^d$ for some values of $s_i$. We obtain our results using the method of Bellman functions.

Adaptive testing on a regression function at a point

We consider the problem of inference on a regression function at a point when the entire function satisfies a sign or shape restriction under the null. We propose a test that achieves the optimal minimax rate adaptively over a range of H\"{o}lder classes, up to a $\log\log n$ term, which we show to be necessary for adaptation. We apply the results to adaptive one-sided tests for the regression discontinuity parameter under a monotonicity restriction, the value of a monotone regression function at the boundary and the proportion of true null hypotheses in a multiple testing problem.

On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions

We give relatively simple sufficient conditions on a Fourier multiplier, so that it maps functions with the H$\ddot{o}$lder property with respect to a part of the variables to functions with the H$o$lder property with respect to all variables. With the using of these sufficient conditions we prove the solvability in H$\ddot{o}$lder classes of the initial-boundary value problems for the linearized Cahn-Hilliard equation with dynamic boundary conditions of two types. For the solutions of these problems Schauder estimates are obtained. The final expanded version of this paper is available at AIMS Journals, Evolution Equations and Control Theory (EECT) at http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=11870

Constructive Characteristic of ho¨ Lder Classes and M-Term Approximations in the Multiple Haar Basis

In terms of the best polynomial approximations in the multiple Haar basis, we obtain a constructive characteristic of the Holder classes H of functions defined on the unit cube $$ \mathbb{I} $$ d of the space ℝ d under the restriction $$ 0<\alpha <\frac{1}{p}\le 1 $$ . We also solve the problem of order estimates of the best m-term approximations in the Haar basis of classes H in the Lebesgue spaces L q ( $$ \mathbb{I} $$ d ).

Classical solvability of multidimensional two-phase Stefan problem for degenerate parabolic equations and Schauder’s estimates for a degenerate parabolic problem with dynamic boundary conditions

We prove locally in time the existence of a smooth solution for multidimensional two-phase Stefan problem for degenerate parabolic equations of the porous medium type. We establish also natural H\"{o}lder class for the boundary conditions in the Cauchy-Dirichlet problem for a degenerate parabolic equation.

Weighted Endpoint Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operators

Let be a Schr&#xf6;dinger operator, where is the laplacian on and the nonnegative potential belongs to the reverse H&#xf6;lder class for some . Assume that . Denote by the weighted Hardy space related to the Schr&#xf6;dinger operator . Let be the commutator generated by a function and the Riesz transform . Firstly, we show that the operator is bounded from into . Secondly, we obtain the endpoint estimates for the commutator . Namely, it is bounded from the weighted Hardy space into .

On adaptive inference and confidence bands

The problem of existence of adaptive confidence bands for an unknown density $f$ that belongs to a nested scale of H\"{o}lder classes over $\mathbb{R}$ or $[0,1]$ is considered. Whereas honest adaptive inference in this problem is impossible already for a pair of H\"{o}lder balls $\Sigma(r),\Sigma(s),r\ne s$, of fixed radius, a nonparametric distinguishability condition is introduced under which adaptive confidence bands can be shown to exist. It is further shown that this condition is necessary and sufficient for the existence of honest asymptotic confidence bands, and that it is strictly weaker than similar analytic conditions recently employed in Gin\'{e} and Nickl [Ann. Statist. 38 (2010) 1122--1170]. The exceptional sets for which honest inference is not possible have vanishingly small probability under natural priors on H\"{o}lder balls $\Sigma(s)$. If no upper bound for the radius of the H\"{o}lder balls is known, a price for adaptation has to be paid, and near-optimal adaptation is possible for standard procedures. The implications of these findings for a general theory of adaptive inference are discussed.

Rates of convergence for the posterior distributions of mixtures of Betas and adaptive nonparametric estimation of the density

In this paper, we investigate the asymptotic properties of nonparametric Bayesian mixtures of Betas for estimating a smooth density on $[0,1]$. We consider a parametrization of Beta distributions in terms of mean and scale parameters and construct a mixture of these Betas in the mean parameter, while putting a prior on this scaling parameter. We prove that such Bayesian nonparametric models have good frequentist asymptotic properties. We determine the posterior rate of concentration around the true density and prove that it is the minimax rate of concentration when the true density belongs to a H\"{o}lder class with regularity $\beta$, for all positive $\beta$, leading to a minimax adaptive estimating procedure of the density. We also believe that the approximating results obtained on these mixtures of Beta densities can be of interest in a frequentist framework.

H1 BOUNDEDNESS FOR RIESZ TRANSFORM RELATED TO SCHRO¨ DINGER OPERATOR ON NILPOTENT GROUPS

Let $\gg$ be a nilpotent Lie groups equipped with a H\{o}rmander system of vector fields $X=(X_1,\ldots,X_m)$ and $\Delta=\sum_{i=1}^mX_i^2$ be the sub-Laplacians associated with $X$. Let $A=-\Delta +W$ be the Schr\{o}dinger operator with the potential function $W$ belongs to the reverse H\{o}lder class $B_q$ for some $q\ge D/2$, where $D$ denote the dimension at infinity. In this paper, we prove that the Riesz transform $\nabla A^{-1/2}$ related to Schr\{o}dinger operator $A$ is bounded from the Hardy space $H^1(\gg)$ to itself.

Lp boundedness of Riesz transform related to Schroedinger operators on a manifold

We establish various $L^{p}$ estimates for the Schrodinger operator $-\Delta+V$ on Riemannian manifolds satisfying the doubling property and a Poincare inequality, where $\Delta $ is the Laplace-Beltrami operator and $V$ belongs to a reverse H\{o}lder class. At the end of this paper we apply our result to Lie groups with polynomial growth.

Tauberian type theorem for operators with interpolation spectrum for Hölder classes

We consider an invertible operator $T$ on a Banach space $X$ whose spectrum is an interpolating set for Hölder classes. We show that if $\|T^{n}\|=O(n^p)$, $p\geq 1$, $\|T^{-n}\|=O(w_n)$ with $n^q=o(w_n)$ $\forall q\in \mathbb {N}$ and $\sum _n 1/(n^{1-\alpha } (\log w_{n})^{1+\alpha })=+\infty$, then $\|T^{-n}\|=O(n^{p+s})$ for all $s > \tfrac {1}{2}$, assuming that $(w_n)_{n\geq 1}$ satisfies suitable regularity conditions. When $X$ is a Hilbert space and $p=0$ (i.e. $T$ is a contraction), we show that under the same assumptions, $T$ is unitary and this is sharp.

Intersections of continuous, Lipschitz, Hölder class, and smooth functions

We present some improvements of known theorems and examples concerning intersections of continuous or Lipschitz functions with smooth functions or intersections of smooth functions or Hölder class functions with smoother functions. We are particularly concerned with our ability to force the projection of the intersection to be uncountable within a given set M which is either large in measure or in category (or both).

Solvability of a certain inverse problem for a parabolic equation in a H�lder class

Solvability of a certain inverse problem for a parabolic equation in a H�lder class

Constructive description of analytic functions of the H�lder class

Constructive description of analytic functions of the H�lder class

Peak sets for analytic H�lder classes

A closed subset E of the unit circumference T is said to be a peak set for the analytic Holder class Aα, 0 < α < 1 there exists a functionf,f∈Aα such that f¦E≡1 and ¦f(z)¦<1 for . It is shown that the set E is a peak set of the algebra Aα if and only if there exists a nonnegative Borel measure μ on T such that the function coincides almost everywhere with a function of the Holder class ⋀α, equal to zero on E. A sufficient condition in order that a closed set E should belong to the family of peak sets is obtained.

Peak sets for H�lder classes (a counterexample to E. M. Dyn'kin's conjecture)

One constructs a peak set E⊂T of the analytic Hölder class Aα [where α is any number from the interval (0, 1)], for which the functiont↦dist(t, E)−α is not summable.