This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region Ω ⊂ R . The inequalities bound (semi-)norms of the boundary trace by certain norms of the function and its gradient on the region and two specific constants kρ and kΩ associated with the domain and a weight function. These inequalities are sharp in that there are functions for which eq...

We present a mixed method for a three-dimensional axisymmetric div-curl system reduced to a two-dimensional computational domain via cylindrical coordinates. We show that when the meridian axisymmetric Maxwell problem is approximated by a mixed method using the lowest order Nédélec elements (for the vector variable) and linear elements (for the Lagrange multiplier), one obtains optimal error es...

In this paper we introduce and develop the notion of minimal subspaces in the framework of algebraic and topological tensor product spaces. This mathematical structure arises in a natural way in the study of tensor representations. We use minimal subspaces to prove the existence of a best approximation, for any element in a Banach tensor space, by means a tensor given in a typical representatio...

We derive bounds on the error, in high-order Sobolev norms, incurred approximation of Sobolev-regular as well analytic functions by neural networks with hyperbolic tangent activation function. These provide explicit estimates error respect to size networks. show that tanh only two hidden layers suffice approximate at comparable or better rates than much deeper ReLU

In this paper, it is constructed that a semi-orthogonal cubic spline wavelet basis of homogeneous Sobolev space H 2 0 (I), which turns out to be a basis of continuous space C 0 (I). Meanwhile, the orthogonal projections on the wavelet subspaces in H 2 0 (I) are extended to the interpolating operators on the corresponding wavelet subspaces in C 0 (I). It is also given that a fast discrete wavele...

Let Jα k be a real power of the integration operator Jk defined on Sobolev space W k p [0, 1]. We investigate the spectral properties of the operator Ak = ⊕n j=1 λjJ α k defined on ⊕n j=1W k p [0, 1]. Namely, we describe the commutant {Ak} ′, the double commutant {Ak} ′′ and the algebra AlgAk. Moreover, we describe the lattices LatAk and HypLatAk of invariant and hyperinvariant subspaces of Ak,...

Let H = −∆ + V (x) be a Schrödinger operator on R, m ≥ 1, with real potential V (x) such that |V (x)| ≤ C〈x〉, 〈x〉 = (1+ |x|2) 1 2 , for some δ > 2. Then, H with domain D(H) = H(R), the Sobolev space of order 2, is selfadjoint in the Hilbert space H = L(R) and C 0 (R) is a core. The spectrum σ(H) ofH consists of absolutely continuous part [0,∞) and a finite number of non-positive eigenvalues {λj...