Journal of Approximation Theory最新文献

Singular and oscillatory functions play a crucial role in various applications, and their approximation is crucial for solving applied mathematics problems efficiently. Hyperinterpolation is a discrete projection method approximating functions with the $L^2$ orthogonal projection coefficients obtained by numerical integration. However, this approach may be inefficient for approximating singular and oscillatory functions, requiring a large number of integration points to achieve satisfactory accuracy. To address this issue, we propose a new approximation scheme in this paper, called efficient hyperinterpolation, which leverages the product-integration methods to attain the desired accuracy with fewer numerical integration points than the original scheme. We provide theorems that explain the superiority of efficient hyperinterpolation over the original scheme in approximating such functions belonging to $L^1$, $L^2$, and continuous function spaces, respectively, and demonstrate through numerical experiments on the interval and the sphere that our approach outperforms the original method in terms of accuracy when using a limited number of integration points.

Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a foundation of current science. In this paper, we find a new integral representation of the Whittaker function of the first kind and show a relevant summation formula for Kummer’s confluent hypergeometric functions. We also perform the specifications of our identities to link to known and new results.

In this paper we study in detail some characterizations of Korovkin closures and we also introduce the notions of onesided upper and lower Korovkin closures. We provide some complete characterizations of these new closures which separate the roles of approximating functions in a Korovkin system. We also present some new characterizations of the classical Korovkin closure in spaces of integrable functions. Again we can introduce and characterize the upper and lower Korovkin closures. Finally, we provide some examples which justify the interest in these new closures.

We study approximative and geometric properties of Chebyshev sets composed of at most countably many planes (i.e., closed affine subspaces). We will assume that the union of planes is irreducible, i.e., no plane in this union contains another plane from the union. We show, in particular, that if a Chebyshev subset $M$ of a Banach space $X$ consists of at least two planes, then it is not $B$-connected (i.e., its intersection with some closed ball is disconnected) and is not $\mathring{B}$-complete. We also verify that, in reflexive $\left(\mathrm{CLUR}\right)$-spaces (and, in particularly, in complete uniformly convex spaces), a set composed of countably many planes is not a Chebyshev set. For finite unions, we show that any finite union of planes (involving at least two planes) is not a Chebyshev set for any norm on the space. Several applications of our results in the spaces $C\left(Q\right)$, $L^1$ and $L^{\infty }$ are also given.

The purpose of this paper is the study of the rate of convergence of Poisson integrals for Laguerre expansions. The convergence of partial sums of Fourier series of functions in $L^p$ spaces was studied, for several classes of orthogonal polynomials. In the Laguerre case Askey and Waigner proved convergence for functions $f\in L^p\left(0,+\infty \right)$ with $4/3<p<4$. In this paper we deal with the Poisson integral $A_{r}f$$\left(0<r<1\right)$ which arises by applying Abel’s summation method to the Laguerre expansion of the function $f$. About 50 years ago, Muckenhoupt intensively studied the Poisson integral for the Laguerre and Hermite polynomials. Among other things he proved pointwise convergence, the convergence by norm, and that the Poisson integral is a contraction mapping in $L^p\left(0,\infty \right)$. Toczek and Wachnicki gave a Voronovskaja-type theorem by calculating the limit ${\left(1-r\right)}^{-1}\left(\left(A_{r}f\right)\left(x\right)-f\left(x\right)\right)$ as $r\to 1^{-}$, provided that $f^{\prime \prime }\left(x\right)$ exists. We generalize this formula by deriving a complete asymptotic development. All its coefficients are explicitly given in a concise form. As an application we apply extrapolation methods in order to improve the rate of convergence of $\left(A_{r}f\right)\left(x\right)$ as $r\to 1^{-}$.

Rubio de Francia proved the one-sided version of Littlewood–Paley inequality for arbitrary intervals. In this paper, we prove the similar inequality in the context of arbitrary Vilenkin systems (that is, for functions on infinite products of cyclic groups). There are no assumptions on the orders of these groups.

We obtain the sharp Kolmogorov inequality ${\Vert y^{\prime }\Vert }_{L_2\left(G\right)}\le \frac{2\sqrt{2}}{\sqrt{3}}{\Vert y\Vert }_{L_1\left(G\right)}^{1/2}{\Vert y_{+}^{\prime \prime }\Vert }_{L_{\infty }\left(G\right)}^{1/2}$ on the real line $G=R$ and the period $G=[0,1)$.

In this paper, we consider the coefficient-based regularized distribution regression which aims to regress from probability measures to real-valued responses over a reproducing kernel Hilbert space (RKHS), where the regularization is put on the coefficients and kernels are assumed to be indefinite. The algorithm involves two stages of sampling, the first stage sample consists of distributions and the second stage sample is obtained from these distributions. The asymptotic behavior of the algorithm is comprehensively studied across different regularity ranges of the regression function. Explicit learning rates are derived by using kernel mean embedding and integral operator techniques. We obtain the optimal rates under some mild conditions, which match the one-stage sampled minimax optimal rate. Compared with the kernel methods for distribution regression in existing literature, the algorithm under consideration does not require the kernel to be symmetric or positive semi-definite and hence provides a simple paradigm for designing indefinite kernel methods, which enriches the theme of the distribution regression. To the best of our knowledge, this is the first result for distribution regression with indefinite kernels, and our algorithm can improve the learning performance against saturation effect.

We study the stability of Triebel–Lizorkin regularity of bounded functions and Lipschitz functions under bi-Lipschitz changes of variables and the regularity of the inverse function of a Triebel–Lizorkin bi-Lipschitz map in Lipschitz domains. To obtain the results we provide an equivalent norm for the Triebel–Lizorkin spaces with fractional smoothness in uniform domains in terms of the first-order difference of the last weak derivative available averaged on balls.

We prove mean convergence of the Fourier series in Akhiezer–Chebyshev polynomials in $L^p$, $p>1$, using a weighted inequality for the Hilbert transform in an arc of the unit circle.