scholarly journals Characterization of functions in terms of rate of convergence of a quadrature process. II. Case of non-periodic functions

1992 ◽  
Vol 163 (1) ◽  
pp. 168-183
Author(s):  
Q.I Rahman ◽  
G Schmeisser
Keyword(s):  
Rate Of Convergence ◽  
Periodic Functions

scholarly journals Hecke’s modular forms by functional equations

2018 ◽  
Vol 14 (05) ◽  
pp. 1247-1256
Author(s):  
Bernhard Heim
Keyword(s):  
Modular Forms ◽  
Functional Equations ◽  
Hecke Operators ◽  
General Context ◽  
Periodic Functions

We investigate the interplay between multiplicative Hecke operators, including bad primes, and the characterization of modular forms studied by Hecke. The operators are applied on periodic functions, which lead to functional equations characterizing certain eta-quotients. This can be considered as a prototype for functional equations in the more general context of Borcherds products.

scholarly journals Rate of convergence for periodic homogenization of convex Hamilton–Jacobi equations in one dimension

2021 ◽  
Vol 121 (2) ◽  
pp. 171-194
Author(s):  
Son N.T. Tu
Keyword(s):  
Rate Of Convergence ◽  
Classical Mechanics ◽  
One Dimension ◽  
Separable Potential ◽  
Periodic Homogenization ◽  
Periodic Functions ◽  
Effective Equation ◽  
Optimal Rate Of Convergence ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

Let u ε and u be viscosity solutions of the oscillatory Hamilton–Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence O ( ε ) of u ε → u as ε → 0 + for a large class of convex Hamiltonians H ( x , y , p ) in one dimension. This class includes the Hamiltonians from classical mechanics with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for periodic functions in dimension n = 1.

scholarly journals Convergence Rates in the Implicit Renewal Theorem on Trees

2013 ◽  
Vol 50 (4) ◽  
pp. 1077-1088
Author(s):  
Predrag R. Jelenković ◽  
Mariana Olvera-Cravioto
Keyword(s):  
Fixed Point ◽  
Rate Of Convergence ◽  
Convergence Rates ◽  
Tail Asymptotics ◽  
Renewal Theorem ◽  
Branching Trees

We consider possibly nonlinear distributional fixed-point equations on weighted branching trees, which include the well-known linear branching recursion. In Jelenković and Olvera-Cravioto (2012), an implicit renewal theorem was developed that enables the characterization of the power-tail asymptotics of the solutions to many equations that fall into this category. In this paper we complement the analysis in our 2012 paper to provide the corresponding rate of convergence.

scholarly journals The degree of approximation by positive operators on compact connected abelian groups

1982 ◽  
Vol 33 (3) ◽  
pp. 364-373 ◽  
Author(s):  
Walter R. Bloom ◽  
Joseph F. Sussich
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Abelian Groups ◽  
Degree Of Approximation ◽  
Linear Operators ◽  
Test Functions ◽  
Periodic Functions ◽  
Quantitative Version ◽  
Approximation By Positive Operators ◽  
Korovkin’S Theorem

AbstractIn 1953 P. P. Korovkin proved that if (Tn) is a sequence of positive linear operators defined on the space C of continuous real 2 π-periodic functions and lim Tnf = f uniformly for f = 1, cos and sin, then lim Tnf = f uniformly for all f ∈ C. Quantitative versions of this result have been given, where the rate of convergence is given in terms of that of the test functions 1, cos and sin, and the modulus of continuity of f. We extend this result by giving a quantitative version of Korovkin's theorem for compact connected abelian groups.

Covering Maps and Periodic Functions on Higher Dimensional Sierpinski Gaskets

2010 ◽  
Vol 61 (5) ◽  
pp. 1151-1181 ◽  
Author(s):  
Huo-Jun Ruan ◽  
Robert S. Strichartz
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Sierpinski Gasket ◽  
Series Expansions ◽  
Periodic Functions ◽  
Sierpiński Gasket ◽  
Higher Dimensional ◽  
Sierpinski Gaskets

Abstract.We construct covering maps from infinite blowups of the$n$-dimensional Sierpinski gasket$S{{G}_{n}}$to certain compact fractafolds based on$S{{G}_{n}}$. These maps are fractal analogs of the usual covering maps fromthe line to the circle. The construction extends work of the second author in the case$n=2$, but a differentmethod of proof is needed, which amounts to solving a Sudoku-type puzzle. We can use the covering maps to define the notion of periodic function on the blowups. We give a characterization of these periodic functions and describe the analog of Fourier series expansions. We study covering maps onto quotient fractalfolds. Finally, we show that such covering maps fail to exist for many other highly symmetric fractals.

On transient behaviour of a nearest-neighbour birth-death process on a lattice

1994 ◽  
Vol 31 (2) ◽  
pp. 549-553 ◽  
Author(s):  
Boris L. Granovsky ◽  
Liat Rozov
Keyword(s):  
Steady State ◽  
Rate Of Convergence ◽  
Death Process ◽  
Nearest Neighbour ◽  
Regular Lattice ◽  
Transient Behaviour ◽  
Coverage Function ◽  
The Mean ◽  
Exact Rate Of Convergence

We provide the explicit expression for the mean coverage function of a generalized voter model on a regular lattice and establish a characterization of the class of the above processes. As a result, we derive the exact rate of convergence of the considered processes to the steady state. We also prove the existence of different processes with the same mean coverage function on a given lattice.

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