Upper Bound for Lebesgue Constant of Bivariate Lagrange Interpolation Polynomial on the Second Kind Chebyshev Points

In the paper, we study the upper bound estimation of the Lebesgue constant of the bivariate Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the second kind on the square − 1,1 2 . And, we prove that the growth order of the Lebesgue constant is O n + 2 2 . This result is different from the Lebesgue constant of Lagrange interpolation polynomial on the unit disk, the growth order of which is O n . And, it is different from the Lebesgue constant of the Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the first kind on the square − 1,1 2 , the growth order of which is O ln n 2 .

Approximation of functions by Complex conformable derivative bases in Frechet spaces

In the present paper the representation, in different domains, of analytic functions by complex conformable fractional derivative bases (CCFDB) and complex conformable fractional integral bases (CCFIB) in Frechet space are investigated . Theorems are proved to show that such representation is possible in closed disks, open disks, open regions surrounding closed disks, at the origin and for all entire functions. Also, some results concerning the growth order and type of CCFDB and CCFIB are determined. Moreover the T-property of CCFDB and CCFIB are dis- cussed. The obtained results recover some known results when alpha = 1. Finally, some applications to the CCFDB and CCFIB of Bernoulli, Euler, Bessel and Chebyshev polynomials have been studied.

Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in C2

This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations f(z+c)2+2αf(z)f(z+c)+f(z)2=eg(z), and the partial differential difference equations f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z12=eg(z),f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z2+∂f(z)∂z1+∂f(z)∂z22=eg(z). We establish some theorems about the forms of the finite order transcendental entire solutions of these functional equations. We also list a series of examples to explain the existence of the finite order transcendental entire solutions of such equations. Meantime, some examples show that there exists a very significant difference with the previous literature on the growth order of the finite order transcendental entire solutions. Our results show that some functional equations can admit the transcendental entire solutions with any positive integer order. These results make a few improvements of the previous theorems given by Xu and Cao, Liu and Yang.

On the Growth of Higher Order Complex Linear Differential Equations Solutions with Entire and Meromorphic Coefficients

We revisit the problem of studying the solutions growth order in complex higher order linear differential equations with entire and meromorphic coefficients of p,q-order, proving how it is related to the growth of the coefficient of the unknown function under adequate assumptions. Our study improves the previous results due to J. Liu - J. Tu - L. Z Shi, L.M. Li - T.B. Cao, and others.

On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth

Suppose {(M^{n},g)} is a Riemannian manifold with nonnegative Ricci curvature, and let {h_{d}(M)} be the dimension of the space of harmonic functions with polynomial growth of growth order at most d. Colding and Minicozzi proved that {h_{d}(M)} is finite. Later on, there are many researches which give better estimates of {h_{d}(M)}. In this paper, we study the behavior of {h_{d}(M)} when d is large. More precisely, suppose {(M^{n},g)} has maximal volume growth and has a unique tangent cone at infinity. Then when d is sufficiently large, we obtain some estimates of {h_{d}(M)} in terms of the growth order d, the dimension n and the asymptotic volume ratio {\alpha=\lim_{R\rightarrow\infty}\frac{\mathrm{Vol}(B_{p}(R))}{R^{n}}}. When {\alpha=\omega_{n}}, i.e., {(M^{n},g)} is isometric to the Euclidean space, the asymptotic behavior obtained in this paper recovers a well-known asymptotic property of {h_{d}(\mathbb{R}^{n})}.

The Growth of Entire Functions Defined by Laplace–Stieltjes Transforms Related to Proximate Order and Approximation

In this article, we discuss the growth of entire functions represented by Laplace–Stieltjes transform converges on the whole complex plane and obtain some equivalence conditions about proximate growth of Laplace–Stieltjes transforms with finite order and infinite order. In addition, we also investigate the approximation of Laplace–Stieltjes transform with the proximate order and obtain some results containing the proximate growth order, the error, An∗, and λn, which are the extension and improvement of the previous theorems given by Luo and Kong and Singhal and Srivastava.

On the Growth Order and Growth Type of Entire Functions of Several Complex Matrices

In this paper, we establish an explicit relation between the growth of the maximum modulus and the Taylor coefficients of entire functions in several complex matrix variables (FSCMVs) in hyperspherical regions. The obtained formulas enable us to compute the growth order and the growth type of some higher dimensional generalizations of the exponential, trigonometric, and some special FSCMVs which are analytic in some extended hyperspherical domains. Furthermore, a result concerning linear substitution of the mode of increase of FSCMVs is given.

Orthorecursive expansion of unity

We study the properties of a sequence [Formula: see text] defined by the recursive relation [Formula: see text] for [Formula: see text] and [Formula: see text]. This sequence also has an alternative definition in terms of certain norm minimization in the space [Formula: see text]. We prove estimates on the growth order of [Formula: see text] and the sequence of its partial sums, infinite series identities, connecting [Formula: see text] with the harmonic numbers [Formula: see text] and also formulate some conjectures based on numerical computations.