Mathematics

This paper concerns the universal approximation property with neural networks in variable Lebesgue spaces. We show that, whenever the exponent function of the space is bounded, every function can be approximated with shallow neural networks with any desired accuracy. This result subsequently leads to determine the universality of the approximation depending on the boundedness of the exponent function. Furthermore, whenever the exponent is unbounded, we obtain some characterization results for the subspace of functions that can be approximated.

We extend the Gamma_2 calculus of Bakry and Emery to include a Carre du champ operator with multiplicative term, providing results which allow to analyse inhomogeneous diffusions.

We shall say that a densely defined closed operator T on a Hilbert space is balanced if $\cD(T)=\cD(T^*)$. Balanced operators are described in terms of their phase operators abnd their moduli. Examples of balanced operators are developed. A characterization of the domain equality $\cD(A)=\cD(B)$ for positive self-adjoint operators A and B with bounded inverses is given in terms of their spectral measures.

In this article, we prove the proximinality of closed unit ball of M -ideals of compact operators. We also prove the ball proximinality of M -embedded spaces in their biduals. Moreover, we show that K( ℓ 1 ) , the space of compact operators on ℓ 1 , is ball proximinal in B( ℓ 1 ) , the space of bounded operators on ℓ 1 , even though K( ℓ 1 ) is not an M -ideal in B( ℓ 1 ) .

Commutator methods are applied to get limiting absorption principles for the discrete standard and Molchanov-Vainberg Schrödinger operators H std =?+V and H MV =D+V on ??2 ( Z d ) , with emphasis on d=1,2,3 . Considered are electric potentials V satisfying a long range condition of the type: V??? κ j V decays appropriately for some κ?�N and all 1?�j?�d , where ? κ j V is the potential shifted by κ units on the j th coordinate. More comprehensive results are obtained for specific small values of κ , such as κ=1,2,3,4 . In this article, we work in a simplified framework in which the main takeaway appears to be the existence of bands where a limiting absorption principle holds, and hence absolutely continuous (a.c.) spectrum, for κ>1 and ? (resp.\ κ>2 and D ). Other decay conditions for V arise from an isomorphism between ? and D in dimension 2. Oscillating potentials are natural examples in application.

Let X be a rearrangement-invariant space over a non-atomic σ -finite measure space (R,μ) and let α∈(0,∞) . We define the functional ∥f ∥ X ⟨α⟩ =∥((|f | α ) ∗∗ ) 1 α ∥ X ¯ ¯ ¯ ¯ ¯ (0,μ(R)) , in which f is a μ -measurable scalar function defined on (R,μ) and X ¯ ¯ ¯ ¯ (0,μ(R)) is the representation space of X . We denote by X ⟨α⟩ the collection of all almost everywhere finite functions f such that ∥f ∥ X ⟨α⟩ is finite. These spaces recently surfaced in connection of optimality of target function spaces in general Sobolev embeddings involving upper Ahlfors regular measures. We present a variety of results on these spaces including their basic functional properties, their relations to customary function spaces and mutual embeddings and, in a particular situation, a characterization of their associate structures. We discover a new one-parameter path of function spaces leading from a Lebesgue space to a Zygmund class and we compare it to the classical one.

We define bidual bounded uo -convergence in vector lattices and investigate relations between this convergence and b -property. We prove that for a regular Riesz dual system ⟨X, X ∼ ⟩ , X has b -property if and only if the order convergence in X agrees with the order convergence in X ∼∼ .

We investigate a generalization of Binet's factorial series in the parameter α μ(z)= ??m=1 ??b m (α) ??m?? k=0 (z+α+k) for the Binet function μ(z)=log?(z)??z??1 2)logz+z??1 2 log(2?) After a brief review of the Binet function μ(z) , several properties of the Binet polynomials b m (α) are presented. We compute the corresponding factorial series for the derivatives of the Binet function and apply those series to the digamma and polygamma functions. Finally, we compare Binet's generalized factorial series with Stirling's \emph{asymptotic} expansion and demonstrate by a numerical example that, with a same number of terms evaluated, the Binet generalized factorial series with an optimized value of α can beat the best possible accuracy of Stirling's expansion.

Motivated by the geodesic barycenter problem from optimal transportation theory, we prove a natural generalization of the Blaschke-Santalo inequality and the affine isoperimetric inequalities for many sets and many functions. We derive from it an entropy bound for the total Kantorovich cost appearing in the barycenter problem. We also establish a "pointwise Prekopa-Leindler inequality" and show a monotonicity property of the multimarginal Blaschke-Santalo functional.

Let S be a symmetric linear relation in the Pontyagin space (K,[.,.]) and let ?=(H, ? 0 , ? 1 ) be the corresponding boundary triple. We prove that the corresponding Weyl function Q satisfies Q??N κ (H) . Conversely, for regular Q??N κ (H) , we find linear relation S?�A , where A is representing self-adjoint linear relation of Q , and we prove that Q is the Weyl function of the relation S . We also prove A ^ =ker ? 1 , where A ^ is the representing relation of the Q ^ :=??Q ?? . In addition, if we assume that the derivative at infinity Q ??(??:= lim z?��? zQ(z) is a boundedly invertible operator then we are able to decompose A , A ^ and S + in terms of S , i.e. we express relation matrices of A , A ^ and S + in terms of S , which is a bounded operator in this case.