Motivated by mathematical learning from training data, we introduce the notion of refinable kernels. Various characterizations of refinable kernels are presented. The concept of refinable kernels leads to the introduction of wavelet-like reproducing kernels. We also investigate a refinable kernel that forms a Riesz basis. In particular, we characterize refinable translation invariant kernels, a...

In this article, a new biorthogonal multiwavelet basis on the interval with complementary homogeneous Dirichlet boundary conditions of second order is presented. This construction is based on the multiresolution analysis onR introduced in [DHJK00] which consists of cubic Hermite splines on the primal side. Numerical results are given for the Riesz constants and both a non-adaptive and an adapti...

This paper considers the controllability and Riesz basis generation property of linear infinite dimensional systems with group generators and one-dimensional admissible input operators. The corresponding results of [Advances in Mathematical Systems Theory, 2000, pp. 221-242] under the assumption of algebraic simplicity for eigenvalues of the generator are generalized to the case in which the ei...

We study the exact boundary controllability of two coupled one dimensional wave equations with a control acting only in one equation. The problem is transformed into a moment problem. This framework has been used in control theory of distributed parameter systems since the classical works of A.G. Butkovsky, H.O. Fattorini and D.L. Russell in the late 1960s to the early 1970s. We use recent resu...

It is well known that a shift invariant space S(N) generated by a compactly supported function N whose integer translates are a Riesz basis of S(N) has a unique orthonormal basis generated by a fundamental function of exponential decay. This result is extended by showing how the localisation carries over to the error of the orthogonal approximation P f] from S(N), i.e. localised error bounds ar...

A Laplace type boundary value problem is considered with a generally discontinuous diffusion coefficient. A domain decomposition technique is used to construct a piecewise tensor product wavelet basis that, when normalised w.r.t. the energy-norm, has Riesz constants that are bounded uniformly in the jumps. An adaptive wavelet Galerkin method is applied to solve the boundary value problem with t...

In this paper we use the standard terminology and notations of the Riesz spaces theory (see [2]). The Banach lattice of the continuous functions from a compact Hausdorff space into a Banach lattice E is denoted by C(K,E). If E = R then we write C(K) instead of C(K,E). 1 stands for the unit function in C(K). One version of the Banach–Stone theorem states that: Theorem 1. Let X and Y be compact H...

To represent the local orientation and energy of a 1-D image signal, many models of early visual processing employ bandpass quadrature filters, formed by combining the original signal with its Hilbert transform. However, representations capable of estimating an image signal's 2-D phase have been largely ignored. Here, we consider 2-D phase representations using a method based upon the Riesz tra...

The exact controllability to the origin for linear evolution control equation is considered. The problem is investigated by its transformation to infinite linear moment problem of generalized exponentials. The existence of solutions of obtained moment problem is investigated for the case when exponentials of a moment problem do not constitute a Riesz basis. The exact controllability of linear c...

We characterize Riesz frames and prove that every Riesz frame is the union of a finite number of Riesz sequences. Furthermore, it is shown that for piecewise continuous wavelets with compact support, the associated regular wavelet systems can be decomposed into a finite number of linearly independent sets. Finally, for finite sets an equivalent condition for decomposition into a given number of...