Knowlpedia

Definition

Small divisor

A small nonzero spectral denominator that amplifies coefficients when a linear equation is inverted.

Suppose a diagonal equation reads λmu^(m)=f^(m)\lambda_m\widehat u(m)=\widehat f(m). A small divisor is a nonzero denominator λm\lambda_m whose small size makes

u^(m)=f^(m)/λm\widehat u(m)=\widehat f(m)/\lambda_m

large relative to the input coefficient. A small-divisor problem concerns controlling these inverses across the frequency set.

Directional derivatives

For v⋅∇v\cdot\nabla, the denominators are 2πi(v⋅m)2\pi i(v\cdot m). Exact zeros are requiring compatibility. Nonzero denominators bounded below by a reciprocal polynomial yield at most polynomial coefficient growth under inversion; faster decay of denominators can cause more severe regularity loss. Nonvanishing alone is not a uniform estimate.