Given a finite subset Ξ ⊂ R and data |Ξ , the surface spline interpolant to the data |Ξ is a function s which minimizes a certain seminorm subject to the interpolation conditions |Ξ = |Ξ . It is known that surface spline interpolation is stable on the Sobolev space W in the sense that ‖s‖L∞(Ω) ≤ const ‖f‖Wm , where m is an integer parameter which specifies the surface spline. In this note we sh...

Let \(\mathcal {H}\) be a complex Hilbert space and let A positive operator on {H}\). We obtain new bounds for the A-numerical radius of operators in semi-Hilbertian {B}_A(\mathcal {H})\) that generalize improve existing ones. Further, we estimate an upper bound \(\mathbb {A}\)-operator seminorm \(2\times 2\) matrices, where {A}=\text{ diag }(A,A)\). The obtained here generalizes earlier relate...

We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods, specifically by using the BV seminorm. Although our procedure applies in quite general situations it was obtained by geometric...

Abstract In the case of radiography of a cylindrically symmetric object, the Abel transform is useful for describing the tomographic measurement operator. The inverse of this operator is unbounded, so regularization is required for the computation of satisfactory inversions. We introduce the use of the total variation seminorm for this purpose, and prove the existence and uniqueness of solution...

We follow the notation of [7,10]. In particular, if E[τ ] is a locally convex space (in short l.c.s.), σ(E, E∗), μ(E,E∗) and β(E,E∗) will denote, respectively, the weak, Mackey and strong topology corresponding to the dual pair 〈E, E∗〉. If U is a neighbourhood of 0 in E[τ ], EU denotes the Banach space associated to U and φU : E → EU denotes the corresponding quotient map. Definition 1. Let E[τ...

A new fractional order seminorm, ICTV r , r ∈ R , r ≥ 1 , is proposed in the one-dimensional setting, as a generalization of the standard ICTV k -seminorms, k ∈ N . The fractional ICTV r -seminorms are shown to be intermediate between the standard ICTV k -seminorms of integer order. A bilevel learning scheme is proposed, where under a box constraint a simultaneous optimization with respect to t...

In this paper, we propose a technique for building adaptive wavelets by means of an extension of the lifting scheme. Our scheme comprises an adaptive update lifting step and a fixed prediction lifting step. The adaptivity consists hereof that the system can choose between two different update filters, and that this choice is triggered by the local gradient of the original signal. If the gradien...

In contrast to the usual Lipschitz seminorms associated to ordinary metrics on compact spaces, we show by examples that Lipschitz seminorms on possibly non-commutative compact spaces are usually not determined by the restriction of the metric they define on the state space, to the extreme points of the state space. We characterize the Lipschitz norms which are determined by their metric on the ...

In this paper, we investigate a hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac measures. Under assumption that the domain is convex and mesh quasi-uniform, priori error estimate in L 2 -norm proved. By duality argument Oswald interpolation, posteriori estimates errors W 1, p -seminorm are also obtained. Finally, numerical examples provided to validate ...

In this paper, we aim to establish several estimates concerning the generalized Euclidean operator radius of d-tuples A-bounded linear operators acting on a complex Hilbert space H, which leads special case well-known A-numerical for d=1. Here, A is positive H. Some inequalities related A-seminorm are proved. addition, under appropriate conditions, reverse bounds in single and multivariable set...