In this letter, we provide a study of learning in a Hilbert space of vectorvalued functions. We motivate the need for extending learning theory of scalar-valued functions by practical considerations and establish some basic results for learning vector-valued functions that should prove useful in applications. Specifically, we allow an output space Y to be a Hilbert space, and we consider a repr...

We study conical square function estimates for Banach-valued functions, and introduce a vector-valued analogue of the Coifman–Meyer–Stein tent spaces. Following recent work of Auscher–MIntosh–Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator A with certain off-diagonal bounds, such that A always has a bounded H...

Vector logic is a matrix–vector representation of the logical calculus inspired in neural network models. In this algebraic formalism, the truth values map on orthonormal Q-dimensional vectors, the monadic operations are represented by square matrices, and the dyadic operations produce rectangular matrices that act on the Kronecker product of the vector truth values. In this formalism, the theo...

In this paper we investigate the nuclear trace of vector-valued Fourier multipliers on torus and its applications to index theory periodic pseudo-differential operators. First characterise nuclearity operators acting Bochner integrable functions. regards, consider discrete cases. We go address problem finding sharp sufficient conditions for torus. end our investigation with two formulae. First,...

This paper introduces a quasi-interpolation operator for scalarand vector-valued finite element spaces constructed on affine, shape-regular meshes with some continuity across mesh interfaces. This operator gives optimal estimates of the best approximation error in any Lp-norm assuming regularity in the fractional Sobolev spaces W r,p, where p ∈ [1,∞] and the smoothness index r can be arbitraril...

The Navier–Lamé operator of classical elasticity, μ∆v+(λ+μ)∇(∇·v), is the simplest example of a linear differential operator whose second-order terms involve a coupling among the components of a vector-valued function. Similar operators on Riemannian manifolds arise in conformal geometry and in quantum gravity. (In the latter context they have come to be called “nonminimal”, but “exotic” is pro...

We present a method for solving the Stokes problem in unbounded domains. It relies on the coupling of the transparent boundary operator and a spectral method in spherical coordinates. It is done explicitly by the use of vector-valued spherical harmonics. A uniform inf-sup condition is proved, which provides an optimal error estimate.

In this paper it is shown how the vector-valued Laplace transform can be used to obtain existence and uniqueness results for exponentially bounded solutions of abstract linear evolution equations in Banach spaces. The method allows to consider problems for which the characteristic operator is, like many sums, products or limits of closed operators, not necessarily closed or closable.

We introduce two kinds of quasi-inner functions. Since every rationally invariant subspace for a shift operator SK on a vector-valued Hardy space H(Ω, K) is generated by a quasi-inner function, we also provide relationships of quasi-inner functions by comparing rationally invariant subspaces generated by them. Furthermore, we discuss fundamental properties of quasi-inner functions, and quasi-in...