Approximation Theory of Linear Operators

Approximation theory of linear operators investigates the systematic approximation of functions by sequences of linear mappings acting on function spaces. Central to the field are families of positive linear operators—such as Bernstein, Szász, Baskakov and Kantorovich variants—that transform an arbitrary function into simpler polynomial or spline representations. The theory examines convergence properties, rates of approximation and asymptotic behaviour through tools like moduli of continuity, Korovkin’s theorem and Voronovskaya-type results. Extensions to quantum calculus introduce q-analogues of classical operators, enriching the palette of approximation schemes. Recent advances explore nonlinear and max-product formulations, summability methods for series representations and statistical modes of convergence. Applications span numerical integration, solution of differential and integral equations, signal processing and machine learning, where efficient and accurate function reconstruction is paramount.

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Approximation Theory of Linear Operators publication trend

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Technical terms

Linear operator: A mapping between function spaces that preserves addition and scalar multiplication, fundamental to constructing approximation schemes.

Modulus of continuity: A function measuring the maximum variation of a function over small intervals, used to quantify uniform convergence rates.

Korovkin theorem: A criterion providing simple test functions whose convergence under a sequence of positive linear operators guarantees convergence for all continuous functions.

Voronovskaya theorem: A result describing the asymptotic behaviour of approximation errors for certain families of linear operators as the degree tends to infinity.

q-integers: Generalised integer values in quantum calculus that deform classical operators, allowing parameter-dependent convergence analysis.

Statistical convergence: A convergence concept based on the natural density of indices satisfying a prescribed closeness condition, extending classical convergence notions.

References

1. Approximation by nonlinear Meyer-König and Zeller operators based on q-integers. International Journal of Mathematics and Computer in Engineering (2024).
2. Szász-Beta operators via Hermite Polynomial. Journal of King Saud University - Science (2024).
3. A Link between Approximation Theory and Summability Methods via Four-Dimensional Infinite Matrices. Mathematics (2021).