A Unified Framework for Positive Linear Approximation Operators

Provides a unified computational framework for defining, evaluating, analyzing, comparing, and visualizing positive linear approximation operators. The package supports classical and generalized discrete operators, ordinary and central moments, numerical moment verification, approximation error analysis, convergence studies, parameter studies, and user-defined operators. The implemented framework is related to classical and generalized approximation operators described in Szász (1950) , Sucu and Varma (2015) , and Aliaga and Aslan (2026) .

ApproxOp

ApproxOp is an R package providing a unified computational framework for positive linear approximation operators.

The package is designed for defining, evaluating, analyzing, comparing, and visualizing classical and generalized approximation operators within a common interface. It also provides tools for ordinary and central moments, numerical moment verification, approximation errors, convergence analysis, parameter studies, and user-defined operators.

Main features

ApproxOp currently provides:

• a unified family / subfamily / variant operator architecture;
• classical Bernstein, Szasz-Mirakyan, and Baskakov operators;
• Sheffer-type operators, including Appell, Charlier, and Meixner subfamilies;
• ordinary and central moments up to order 8;
• numerical verification of moments;
• pointwise approximation and error analysis;
• convergence studies over different values of n;
• comparison of several approximation operators;
• one- and two-parameter scans;
• graphical tools for approximation, convergence, comparisons, and parameter studies;
• registration and diagnostic checking of user-defined operators.

The package is designed to be extensible, so additional operator families, subfamilies, and variants can be incorporated into the same framework.

Installation

Once available on CRAN, the released version can be installed with:

install.packages("ApproxOp")

The development version can be installed from GitHub with:

# install.packages("pak")
pak::pak("edmond-aliaga/ApproxOp")

Basic example

Load the package:

library(ApproxOp)

Create a Bernstein operator:

B <- approx_operator(
  family = "bernstein",
  n = 20
)

B

Approximate the function (f(x)=x^2):

f <- function(x) x^2

approximate(
  operator = B,
  f = f,
  x = c(0.25, 0.50, 0.75)
)

Moments

Ordinary moments can be computed up to order 8:

moments(
  operator = B,
  order = 4,
  x = 0.5
)

Central moments are obtained with:

moments(
  operator = B,
  order = 4,
  x = 0.5,
  central = TRUE
)

The analytical/computational moment engine can also be checked against direct numerical evaluation:

verify_moments(
  operator = B,
  order = 4,
  x = 0.5
)

Error and convergence analysis

Pointwise approximation errors can be computed with:

approx_error(
  operator = B,
  f = f,
  x = seq(0, 1, length.out = 101)
)

Convergence over several values of n can be studied using:

convergence(
  operator = B,
  f = f,
  grid = seq(0, 1, length.out = 101),
  n = c(5, 10, 20, 50)
)

Comparing operators

ApproxOp provides a common framework for comparing different approximation operators using numerical error measures and graphical tools.

It also supports one- and two-dimensional parameter scans for parameterized operator families.

User-defined operators

New operators can be incorporated through register_operator(). This allows users to extend the framework without modifying the core package.

Registered operators can be examined with:

check_operator(...)

which provides numerical diagnostics such as evaluation, finiteness, constant reproduction, and positivity checks.

Development

ApproxOp is under active development. Future versions may extend the framework with additional operator families and integral or fractional variants.

Bug reports and suggestions can be submitted through the GitHub issue tracker.

Author

Edmond Aliaga

Department of Mathematics and Computer Sciences
University of Prishtina, Kosovo