# IJPAM: Volume 57, No. 5 (2009)

**ON ERROR ESTIMATION FOR APPROXIMATION METHODS**

INVOLVING DOMAIN DISCRETIZATION

I: PROBLEM SETTING

INVOLVING DOMAIN DISCRETIZATION

I: PROBLEM SETTING

Priority R&D Group for Mathematical Modelling

Numerical Simulation and Computer Visualization

Narvik University College

2, Lodve Lange's St., P.O. Box 385, N-8505 Narvik, NORWAY

e-mail: ltd@hin.no

url: https://ansatte.hin.no/ltd/

**Abstract.**This is the first of a sequence of 12 papers, including also
[citetions] (in this order), dedicated to
the study of error estimates for approximation problems based on
discretization of the domain of the approximated functions, and (in
the concluding paper [citation]) comparison of the similarities
and differences with the error estimates derived by alternative
approximation methods (typically, of projection type) based on
finite-dimensional subspaces of functions having the same continual
domain as the target function. This paper is dedicated to the
following:

- In Section 1: indicating general classes of problems in
numerical analysis divided into respective pairs, depending on
whether the definition domain is being discretized into a mesh, or
this domain is continual.
- In Section 2: introducing tools for measuring
the error, such as integral and averaged moduli of smoothness,
related concepts, and comparisons of properties.
- In Section 3: specifying the research objectives
of the research in the entire sequence of papers [citations].
- In Section 4: detailed specification of the model problems to be considered in [citations].

**Received: **May 8, 2009

**AMS Subject Classification: **65L12, 65L70, 65M15, 65M22,
26A15, 26A16, 34A30, 34G05, 34G10, 35G16, 35K05, 35K20, 39A06,
39A14, 39A70, 41A05, 41A25, 41A55, 41A63, 46E35, 46E39, 46N20,
46N30, 46N40, 47B38, 47B39, 47D03, 47D06, 47E05, 47F05, 47N20,
47N30, 47N40, 62D99, 62G08, 65C60, 65D05, 65D10, 65D15, 65D25,
65D30, 65D32, 65J10, 65L10, 65L20, 65L60, 65M05, 65M06, 65M10,
65M12, 65M60, 65N05, 65N06, 65N08, 65N10, 65N12, 65N15, 65N22,
65U05

**Key Words and Phrases: **error, estimate, approximation, step, convergence, rate,
order, domain, mesh, discretization, discrete, semi-discrete,
continual, numerical, analysis, differentiation, integration,
pointwise interpolation, quasi-interpolation, fitting, finite
difference scheme, finite element, non-parametric, regression,
deterministic, indeterministic, random, independent, identically
distributed, bias, variance, sample, knot, risk, modulus of
smoothness, integral, averaged, Riemann sum, uniform, non-uniform,
**K**-functional, Lebesgue space, sequence space, Sobolev space,
Triebel-Lizorkin space, Besov space, interpolation space, real,
complex, **A**-space, Wiener amalgam, metric, norm, quasi-norm,
isomorphism, embedding, bound, equivalence constant, initial,
boundary, initial-boundary, differential equation, ordinary,
partial, stationary, non-stationary, template, functional, linear,
non-linear, positive, univariate, multivariate, multidimensional

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2009

**Volume:** 57

**Issue:** 5