Let T be a self-adjoint tridiagonal operator in a Hilbert space H with the orthonormal basis {e n}n=1∞, σ(T) be the spectrum of T and Λ(T) be the set of all the limit points of eigenvalues of the truncated operator T N. We give sufficient conditions such that the spectrum of T is discrete and σ(T) = Λ(T) and we connect this problem with an old problem in analysis.
The estimate is shown to hold if and only if is elliptic and canceling. Here is a homogeneous linear differential operator of order on from a vector space to a vector space . The operator is defined to be canceling if . This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential...
Multiplicatively invariant (MI) spaces are closed subspaces of L²(Ω, ) that are invariant under multiplication by (some) functions in ; they were first introduced by Bownik and Ross (2014). In this paper we work with MI spaces that are finitely generated. We prove that almost every set of functions constructed by taking linear combinations of the generators of a finitely generated MI space is a new set of generators for the same space, and we give necessary and sufficient conditions on the linear...
The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing our main results, the corresponding integral representations are deduced for such familiar classes of hypergeometric polynomials as (for example) the generalized Bedient polynomials of the first and second kinds. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization...
The extremal functions f0(z) realizing the maxima of some functionals (e.g. max |a3|, and max arg f′(z)) within the so-called universal linearly invariant family Uα (in the sense of Pommerenke [10]) have such a form that f′0(z) looks similar to generating function for Meixner-Pollaczek (MP) polynomials [2], [8]. This fact gives motivation for the definition and study of the generalized Meixner-Pollaczek (GMP) polynomials Pλn (x; θ,ψ) of a real variable x as coefficients of [###] where the parameters...
We present, discuss and apply two reiteration theorems for triples of quasi-Banach function lattices. Some interpolation results for block-Lorentz spaces and triples of weighted -spaces are proved. By using these results and a wavelet theory approach we calculate (θ,q)-spaces for triples of smooth function spaces (such as Besov spaces, Sobolev spaces, etc.). In contrast to the case of couples, for which even the scale of Besov spaces is not stable under interpolation, for triples we obtain stability...
We study continuity envelopes of function spaces and where the weight belongs to the Muckenhoupt class ₁. This essentially extends partial forerunners in [13, 14]. We also indicate some applications of these results.
In the setting of a metric measure space (X, d, μ) with an n-dimensional Radon measure μ, we give a necessary and sufficient condition for the boundedness of Calderón-Zygmund operators associated to the measure μ on Lipschitz spaces on the support of μ. Also, for the Euclidean space Rd with an arbitrary Radon measure μ, we give several characterizations of Lipschitz spaces on the support of μ, Lip(α,μ), in terms of mean oscillations involving μ. This allows us to view the "regular" BMO space of...