相关论文: Best approximation in the mean by analytic and har…
We solve the problem of best approximation by partial isometries of given rank to an arbitrary rectangular matrix, when the distance is measured in any unitarily invariant norm. In the case where the norm is strictly convex, we parametrize…
We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed:…
In a previous paper [Adcock & Huybrechs, 2019] we described the numerical approximation of functions using redundant sets and frames. Redundancy in the function representation offers enormous flexibility compared to using a basis, but…
Approximation of entire functions by their pad\'e approximants has been examined in the past. It is true that generically such an approximation holds. However, examining this problem from another viewpoint, we obtain stronger generic…
Function approximation and recovery via some sampled data have long been studied in a wide array of applied mathematics and statistics fields. Analytic tools, such as the Poincar\'e inequality, have been handy for estimating the…
We revisit the well-established regularity estimates on harmonic maps on surfaces to question their independence with respect to the dimension of the target manifold. We are mainly interested in harmonic maps into target ellipsoids, that we…
Bayesian optimisation is a popular approach for optimising expensive black-box functions. The next location to be evaluated is selected via maximising an acquisition function that balances exploitation and exploration. Gaussian processes,…
This paper studies the probabilistic function approximation problem over reproducing kernel Hilbert spaces. We show the existence and uniqueness of the optimizer under mild assumptions. Furthermore, we generalize the celebrated representer…
The problem of root mean square approximation of a square integrable function by finite linear combinations of exponential functions is considered. It is subdivided into linear and nonlinear parts. The linear approximation problem is…
This is an expository paper on approximating functions from general Hilbert or Banach spaces in the worst case, average case and randomized settings with error measured in the $L_p$ sense. We define the power function as the ratio between…
It is a natural question to ask which plurisubharmonic functions admit a 'nice' approximation in the sense of a decreasing equisingular approximation with analytic singularities. For arbitrary toric plurisubharmonic functions, we give a…
Several concepts of approximate reasoning in uncertainty processing are linked to the processing of distribution functions. In this paper we make use of probabilistic framework of approximate reasoning by proposing a Lebesgue-type approach…
We propose an optimal approximation formula for analytic functions that are defined on a complex region containing the real interval $(-1,1)$ and possibly have algebraic singularities at the endpoints of the interval. As a space of such…
When implementing regular enough functions (e.g., elementary or special functions) on a computing system, we frequently use polynomial approximations. In most cases, the polynomial that best approximates (for a given distance and in a given…
We characterize the best $L_{2}$ approximation to a multivariate function by linear combinations of ridge functions multiplied by some fixed weight functions. In the special case when the weight functions are constants, we propose explicit…
Under investigation is the problem of finding the best approximation of a function in a Hilbert space subject to convex constraints and prescribed nonlinear transformations. We show that in many instances these prescriptions can be…
Best simultaneous approximation (BSA) for finitely or infinitely many functions are considered under the uniform norm and other important norms. Characterization theorems for a BSA from a finite-dimensional subspace are obtained by a…
It is strange but fruitful to think about the functions as random processes. Any function can be viewed as a martingale (in many different ways) with discrete time. But it can be useful to have continuous time too. Processes can emulate…
We consider finite element approximations of ill-posed elliptic problems with conditional stability. The notion of {\emph{optimal error estimates}} is defined including both convergence with respect to mesh parameter and perturbations in…
The problem of estimating the mean of random functions based on discretely sampled data arises naturally in functional data analysis. In this paper, we study optimal estimation of the mean function under both common and independent designs.…