scholarly journals Hermite-Fejér and Grünwald interpolation at generalized Laguerre zeros

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4855-4863
Author(s):  
Bonis De ◽  
David Kubayi
Keyword(s):  
Laguerre Polynomials ◽  
Function Spaces ◽  
Weighted Function ◽  
Weighted Function Spaces ◽  
Generalized Laguerre Polynomials ◽  
Uniformly Convergent

We introduce special Hermite-Fej?r and Gr?nwald operators at the zeros of the generalized Laguerre polynomials. We will prove that these interpolation processes are uniformly convergent in suitable weighted function spaces.

Cesàro summability of lagrange interpolation on Jacobi roots

2004 ◽  
Vol 41 (4) ◽  
pp. 437-451
Author(s):  
L. Szili
Keyword(s):  
Lagrange Interpolation ◽  
Function Spaces ◽  
Cesàro Means ◽  
Cesàro Summability ◽  
Weighted Function ◽  
Weighted Function Spaces ◽  
Cesaro Means ◽  
Cesaro Summability ◽  
Cesáro Means ◽  
Uniformly Convergent

The aim of this paper is to give such weighted function spaces in which the sequence of Cesàro means of Lagrange interpolatory polynomials on Jacobi roots are uniformly convergent.

scholarly journals Fractional Generalizations of Rodrigues-Type Formulas for Laguerre Functions in Function Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 984
Author(s):  
Pedro J. Miana ◽  
Natalia Romero
Keyword(s):  
Integral Representation ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Function Spaces ◽  
Laguerre Functions ◽  
Addition Formula ◽  
Lebesgue Spaces ◽  
Rodrigues Formula ◽  
Generalized Laguerre Polynomials ◽  
New Family

Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them.

scholarly journals Linear Differential Equations on Some Classes of Weighted Function Spaces

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1113
Author(s):  
Ahmed El-Sayed Ahmed ◽  
Amnah E. Shammaky
Keyword(s):  
Specific Growth ◽  
Function Spaces ◽  
Entire Solutions ◽  
Linear Type ◽  
Weighted Function ◽  
Weighted Function Spaces ◽  
Type Classes ◽  
Type Differential Equation ◽  
Linear Differential ◽  
Weighted Type

Some weighted-type classes of holomorphic function spaces were introduced in the current study. Moreover, as an application of the new defined classes, the specific growth of certain entire-solutions of a linear-type differential equation by the use of concerned coefficients of certain analytic-type functions, that is the equation h(k)+Kk−1(υ)h(k−1)+…+K1(υ)h′+K0(υ)h=0, will be discussed in this current research, whereas the considered coefficients K0(υ),…,Kk−1(υ) are holomorphic in the disc ΓR={υ∈C:|υ|<R},0<R≤∞. In addition, some non-trivial specific examples are illustrated to clear the roles of the obtained results with some sharpness sense. Hence, the obtained results are strengthen to some previous interesting results from the literature.

Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients

2019 ◽  
Vol 53 (5) ◽  
pp. 1507-1552 ◽  
Author(s):  
L. Herrmann ◽  
C. Schwab
Keyword(s):  
Monte Carlo ◽  
Random Field ◽  
Gaussian Random Field ◽  
Function Spaces ◽  
Monte Carlo Integration ◽  
Superior Performance ◽  
Elliptic Pdes ◽  
Weighted Function ◽  
Weighted Function Spaces ◽  
Quasi Monte Carlo

We analyze the convergence rate of a multilevel quasi-Monte Carlo (MLQMC) Finite Element Method (FEM) for a scalar diffusion equation with log-Gaussian, isotropic coefficients in a bounded, polytopal domain D ⊂ ℝd. The multilevel algorithm QL* which we analyze here was first proposed, in the case of parametric PDEs with sequences of independent, uniformly distributed parameters in Kuo et al. (Found. Comput. Math. 15 (2015) 411–449). The random coefficient is assumed to admit a representation with locally supported coefficient functions, as arise for example in spline- or multiresolution representations of the input random field. The present analysis builds on and generalizes our single-level analysis in Herrmann and Schwab (Numer. Math. 141 (2019) 63–102). It also extends the MLQMC error analysis in Kuo et al. (Math. Comput. 86 (2017) 2827–2860), to locally supported basis functions in the representation of the Gaussian random field (GRF) in D, and to product weights in QMC integration. In particular, in polytopal domains D ⊂ ℝd, d=2,3, our analysis is based on weighted function spaces to describe solution regularity with respect to the spatial coordinates. These spaces allow GRFs and PDE solutions whose realizations become singular at edges and vertices of D. This allows for non-stationary GRFs whose covariance operators and associated precision operator are fractional powers of elliptic differential operators in D with boundary conditions on ∂D. In the weighted function spaces in D, first order, Lagrangian Finite Elements on regular, locally refined, simplicial triangulations of D yield optimal asymptotic convergence rates. Comparison of the ε-complexity for a class of Matérn-like GRF inputs indicates, for input GRFs with low sample regularity, superior performance of the present MLQMC-FEM with locally supported representation functions over alternative representations, e.g. of Karhunen–Loève type. Our analysis yields general bounds for the ε-complexity of the MLQMC algorithm, uniformly with respect to the dimension of the parameter space.

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