UDC 517.587, 512.643
We consider a Sobolev-type inner product known from the literature. This inner product is a modification of the inner product of Meixner polynomials containing an additional term at a point $a\in\mathbb{R},$ involving the forward-difference operator $\Delta$ and a weight $\lambda>0.$ Here, the point $a$ is chosen to guarantee that the spectrum of Meixner polynomials does not intersect the interval $(a, a+1).$ This inner product and its generalization already exist in the literature for various other measures. We consider a specific measure. For the resulting orthonormal Sobolev-type polynomials, we obtain a five-term recurrence relation. The associated pentadiagonal matrix is analyzed in relation to the Jacobi pencil matrix. The behavior of zeros with respect to the parameters $a$ and $\lambda$ is investigated and the Plancherel–Rotach-type asymptotic behavior is obtained.
Let G be a homogeneous Lie group of dimension ϰ. In this article, we investigate the mapping properties of the Wolff-type potential Wϑ,2 on G, associated with a homogeneous quasi-norm, and show that it coincides, up to an explicit dimensional constant, with the Riesz potential of doubled order 2ϑ on G. Using this ident...
Waqar Afzal, Mujahid Abbas, M. El-Naggar et al.· Fractal and Fractional· 3 citations
We construct and analyze optimal quadrature formulas for the right-sided Riemann–Liouville fractional integral in the Sobolev space L2(m)(t,1), using φ,φ′,…,φ(m−1) at N+1 equally spaced nodes. The analysis is carried out through the Peano kernel of the error functional, for which we give an exact closed form; this yiel...
In this paper, we study local and global properties of positive solutions to the equation $-\Delta u=u^p|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, where $p$ and $q$ are parameters. We introduce a linear operator to construct the differential inequality to obtain gradient estimates, and further establish Liouv...
We prove a surprisingly simple pointwise formula for the Frobenius norm of the tensor of $n$-th order partial derivatives of a radially symmetric function $f(x)=g(r(x))$. Using the iterations of the differential operator $\mathcal{D} g(r)=g'(r)/r$, we avoid technical difficulties usually caused by higher-order radial d...
This paper establishes an asymptotic formula for negative Sobolev norms as the fractional order tends to zero. In the Euclidean setting, under a mild boundedness condition, the product of the order and the norm raised to the power $p$ converges to a dimension-independent constant multiple of the corresponding $L^p$ nor...
In this paper, we relate multiplicities of Bernstein--Sato-type polynomials with respect to a holomorphic function f to several singularity invariants. First, we introduce certain ``mod''b-functions and show that they characterize the weight filtration on the localization of a simple regular holonomic D-module along f,...
A. Lőrincz, Rui-Jie Yang· 3 citations
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