Schwartzvariant

Schwartzvariant is a term used in functional analysis to denote a family of function spaces that generalize or modify the classical Schwartz space S(R^n). The exact definition is not standardized; different authors adopt different variants. In general, a Schwartzvariant space consists of smooth functions whose growth and decay are controlled by a weight function and a system of seminorms. A typical formulation enforces rapid decay of the function and all its derivatives, relative to the weight, so that the space remains stable under differentiation, multiplication by polynomials, and (in many cases) the Fourier transform. By choosing the weight function, one can interpolate between the standard Schwartz space and weighted spaces, producing a spectrum of variants with varying decay rates in different variables or directions.

In most treatments, Schwartzvariant spaces are locally convex and complete, often forming Fréchet or nuclear spaces, which makes them suitable as test function spaces for distributions and for pseudodifferential operator analysis. The variant with weight w ≡ 1 yields the classical Schwartz space. More general weights lead to weighted Schwartzvariant spaces, sometimes related to Gelfand–Shilov or tempered distributions, depending on the level of decay imposed and the smoothness required.

The term is used primarily in specialized texts and seminars on harmonic analysis, microlocal analysis, and PDEs. Because the concept is not standardized, precise properties and inclusion relations with other spaces (such as Lp or Sobolev spaces) vary by definition.

See also: Schwartz space; weighted Schwartz spaces; Gelfand–Shilov spaces; tempered distributions.