Abstract
For functions f1; : : : ; fn on a set D, we characterize their linear
independence with an invertible matrix from their values at n distinct points in
D. With the matrix, the pointwise convergence of a sequence fgkg of functions
in the spanff1; ; fng is shown to be equivalent to those of the sequences
of the coordinates of gks in the span. When fis are bounded, a pointwise
convergent sequence fgkg must uniformly converge to a function in the span.
It turns out that the limit of a convergent sequence fgkg inherits the continuity,
dierentiability, and integrability of fis. Furthermore the (pointwise
or uniform) convergence of a sequence of solutions of an n-th order constant
coecients linear dierential equation is completely determined by that of the
sequence of relevant initial conditions.
independence with an invertible matrix from their values at n distinct points in
D. With the matrix, the pointwise convergence of a sequence fgkg of functions
in the spanff1; ; fng is shown to be equivalent to those of the sequences
of the coordinates of gks in the span. When fis are bounded, a pointwise
convergent sequence fgkg must uniformly converge to a function in the span.
It turns out that the limit of a convergent sequence fgkg inherits the continuity,
dierentiability, and integrability of fis. Furthermore the (pointwise
or uniform) convergence of a sequence of solutions of an n-th order constant
coecients linear dierential equation is completely determined by that of the
sequence of relevant initial conditions.
| Original language | English |
|---|---|
| Pages | 55-68 |
| Number of pages | 14 |
| DOIs | |
| Publication status | Published - 1 Sept 2024 |