Tapproximation

Tapproximation is a framework in approximation theory describing a family of operators {T_t}_{t>0} that approximate a target function as t tends to zero. Each operator T_t maps a function f defined on a domain to another function, often by smoothing or local averaging.

A common form is T_t f(x) = ∫ f(y) K_t(x,y) dy, where K_t is a kernel family with K_t ≥ 0 and ∫ K_t(x,y) dy = 1 for all x. Such families are called approximate identities. Under mild regularity of f (for example, f in Lp or continuous on a compact domain), T_t f converges to f as t → 0. The rate of convergence depends on the regularity of f and the moment or oscillation properties of K_t; for Lipschitz or Hölder continuous f one often has ||T_t f − f|| ≤ C t^α.

Discrete versions include polynomial approximations such as Bernstein polynomials or Fejér means, which can be interpreted as T_t with t related to the degree of the approximants. Connections include semigroups of operators, where t plays the role of time and {T_t} forms a contracting, smoothing family (for example, the Poisson or heat semigroups). Tapproximation also intersects kernel smoothing, radial basis function interpolation, and regularization methods.

Applications of Tapproximation appear in numerical approximation of functions, smoothing of noisy data, and providing theoretical guarantees on convergence and rates in functional and harmonic analysis. The concept encompasses both continuous and discrete settings and serves as a unifying viewpoint for various smoothing and approximation techniques.