A Schwartz function is a C^∞ function that goes to zero as faster than any inverse power of , as do all its derivatives. That is, for all and with nonnegative integer components, there is a real constant such that

where multi-index notation is used for and , and is the standard l²-norm.

The set of all Schwartz functions is called a Schwartz space and is denoted by . The Fourier transform gives a one-to-one correspondence between and itself, taking the pointwise product into the convolution product and vice versa (Hunter 2014, p. 169). With the convention having integral kernel , the Gaussian function is a fixed point of the Fourier transform.

Analogous to , the class consists of functions that go to zero as faster than any inverse power of . Taking Fourier coefficients gives a one-to-one correspondence from onto , where is the -torus, defined as the direct product of copies of the circle .

See also

Convolution, Fourier Transform, Schwartz Space

Portions of this entry contributed by W.D. Van Suijlekom

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References

Gilkey, P. B. Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem. Berkeley, CA: Publish or Perish Press, 1984.Hunter, J. K. Notes on Partial Differential Equations. pp. 166-169, Jun. 18, 2014. https://math.ucdavis.edu/~hunter/pdes/pde_notes.pdf.Richtmyer, R. D. Principles of Advanced Mathematical Physics, Vol. 1. New York: Springer-Verlag, 1978.

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Schwartz Function

Cite this as:

Weisstein, Eric W., with contributions by W.D. Van Suijlekom. "Schwartz Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SchwartzFunction.html

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