We characterize the reproducing kernel Hilbert spaces whose elements are p-integrable functions in terms of the boundedness of the integral operator whose kernel is the reproducing kernel. Moreover, for p = 2 we show that the spectral decomposition of this integral operator gives a complete description of the reproducing kernel.

We study the semi-classical behavior of the spectral function of the Schrödinger operator with short range potential. We prove that the spectral function is a semiclassical Fourier integral operator quantizing the forward and backward Hamiltonian flow relations of the system. Under a certain geometric condition we explicitly compute the phase in an oscillatory integral representation of the spe...

This paper uses frame techniques to characterize the Schatten class properties of integral operators. The main result shows that if the coefficients {〈k,Φm,n〉} of certain frame expansions of the kernel k of an integral operator are in l, then the operator is Schatten p-class. As a corollary, we conclude that if the kernel or Kohn-Nirenberg symbol of a pseudodifferential operator lies in a parti...

The generalized Funk transform is a integral transform acting from densities on a manifold X to functions defined on a family Σ of hypersurfaces in X. The dual operator M is again a Funk transform which is defined for densities on the manifold Σ. We state twoside estimates for the operator M and a description of the range of the Funk transform operatorM and approximation theorem for the kernel ...

A proof of high-order convergence of three deterministic particle methods for the convectiondiffusion equation in two dimensions is presented. The methods are based on discretizations of an integro-differential equation in which an integral operator approximates the diffusion operator. The methods differ in the discretization of this operator. The conditions for convergence imposed on the kerne...

We study finite delay evolution equation { x′(t) = Ax(t) + F (t, xt), t ≥ 0, x0 = φ ∈ C ([−r, 0] , E) , where linear operator A is non-densely defined and satisfies the Hille-Yosida condition. First we obtain some properties of “integral solutions” in this case, and prove the compactness of an operator determined by integral solutions. This allows us to apply Horn’s fixed point theorem to prove...