Error estimates in Sobolev spaces for interpolating thin plate splines under tension

AbstractThis paper discusses Lp-error estimates for interpolation by thin plate spline under tension of a function in the classical Sobolev space on an open bounded set with a Lipschitz-continuous boundary. A property of convergence is also given when the set of interpolating points becomes more and more dense

Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators

In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator $\mathbf{P}$ consisting of finitely or countably many distributional operators $P_n$, which are defined on the dual space of the Schwartz space. The types of operators we consider include not only differential operators, but also more general distributional operators such as pseudo-differential operators. We deduce that a certain appropriate full-space Green function $G$ with respect to $L:=\mathbf{P}^{\ast T}\mathbf{P}$ now becomes a conditionally positive definite function. In order to support this claim we ensure that the distributional adjoint operator $\mathbf{P}^{\ast}$ of $\mathbf{P}$ is well-defined in the distributional sense. Under sufficient conditions, the native space (reproducing-kernel Hilbert space) associated with the Green function $G$ can be isometrically embedded into or even be isometrically equivalent to a generalized Sobolev space. As an application, we take linear combinations of translates of the Green function with possibly added polynomial terms and construct a multivariate minimum-norm interpolant $s_{f,X}$ to data values sampled from an unknown generalized Sobolev function $f$ at data sites located in some set $X \subset \mathbb{R}^d$. We provide several examples, such as Mat\'ern kernels or Gaussian kernels, that illustrate how many reproducing-kernel Hilbert spaces of well-known reproducing kernels are isometrically equivalent to a generalized Sobolev space. These examples further illustrate how we can rescale the Sobolev spaces by the vector distributional operator $\mathbf{P}$. Introducing the notion of scale as part of the definition of a generalized Sobolev space may help us to choose the "best" kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D. thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}

Metabolic constituents of grapevine and grape-derived products

The numerous uses of the grapevine fruit, especially for wine and beverages, have made it one of the most important plants worldwide. The phytochemistry of grapevine is rich in a wide range of compounds. Many of them are renowned for their numerous medicinal uses. The production of grapevine metabolites is highly conditioned by many factors like environment or pathogen attack. Some grapevine phytoalexins have gained a great deal of attention due to their antimicrobial activities, being also involved in the induction of resistance in grapevine against those pathogens. Meanwhile grapevine biotechnology is still evolving, thanks to the technological advance of modern science, and biotechnologists are making huge efforts to produce grapevine cultivars of desired characteristics. In this paper, important metabolites from grapevine and grape derived products like wine will be reviewed with their health promoting effects and their role against certain stress factors in grapevine physiology

PSEUDO-DIFFERENTIAL OPERATOR ASSOCIATED TO THE RADIAL BASIS FUNCTIONS UNDER TENSION.

Abstract. Radial basis functions under tension (RBFT) depend on a positive parameter, incorporate the concept of spline with tension and provide a convenient way for the control of the behavior of the interpolating surface. The RBFT involve a function which is not complicated than exponential and may be easily coded. In this paper, we show that the RBFT, as like thin plate spline, may be associated to a differential operator in a Beppo-Levi space type. Both smoothing and interpolating problems by RBFT are studied. Résumé. Les fonctions splines radiales sous tension dépendent d’un paramètre positif. Ces fonctions permettent d’incorporer un concept de tension pour toute dimension de l’espace. On montre dans ce papier que ces fonctions sont associées à un opérateur pseudo-différentiel dans un espace de type Beppo-Levi et on étudie le problème d’interpolation et de lissage. Des examples numériques sont donnés pour illustrer ce type d’approximation

Blind image restoration as a convex optimization problem

In this paper, we consider the blind image restoration as a convex constrained problem and we propose to solve this problem by a conditional gradient ethod. Such a method is based on a Thikonov regularization technique and is obtained by an approximation of the blur matrix as a Kronecker product of two matrices given as a sum of a Toeplitz and Hankel matrices. Numerical examples are given to show the efficiency of our proposed method