Dunkl

Dunkl refers to a family of mathematical constructs named after Charles F. Dunkl, most notably the Dunkl operators and the Dunkl transform. These objects arise in harmonic analysis and are built to respect the symmetries of finite reflection groups associated with root systems.

Dunkl operators are differential-difference operators associated with a finite reflection group W generated by reflections in a root system R in R^n. For a multiplicity function k defined on the roots, the Dunkl operator T_i acts on smooth functions by combining a partial derivative with a weighted sum over reflections: T_i f(x) = ∂_i f(x) + sum over α in R_+ of k(α) α_i/(α·x) [f(x) - f(s_α x)], where s_α is the reflection through the hyperplane orthogonal to α. The operators T_i commute for all i, enabling a joint spectral theory and a calculus adapted to the symmetry of W.

The Dunkl transform F_k is an integral transform built from the Dunkl kernel E_k(x, y) and generalizes the Fourier transform. It is unitary on L^2 with respect to the weight measure μ_k(dx) = ∏_{α∈R_+} |α·x|^{k(α)} dx and satisfies inversion and Plancherel formulas. The transform reduces to the ordinary Fourier transform when all multiplicities k vanish, and it gives rise to a rich theory of generalized harmonic analysis, including Dunkl harmonics and special functions tied to root systems.

Dunkl theory also intersects probability and mathematical physics, with connections to Dunkl processes and integrable systems. In broader use, the term Dunkl may appear as a surname in various scholarly and professional contexts.