ON UNIVERSAL FUNCTIONS
ON UNIVERSAL FUNCTIONS
An entire function <TEX>$f\;{\in}\;H(\mathbb{C})$</TEX> is called universal with respect to translations if for any <TEX>$g\;{\in}\;H(\mathbb{C}),\;R\;>\;0,\;and\;{\epsilon}\;>\;0$</TEX>, there is <TEX>$n\;{\in}\;{\mathbb{N}}$</TEX> such that <TEX>$<TEX>$\mid$</TEX>f(z\;+\;n)\;-\;g(z)<TEX>$\mid$</TEX>\;<\;{\epsilon}$</TEX> whenever <TEX>$<TEX>$\mid$</TEX>z<TEX>$\mid$</TEX>\;{\leq}\;R$</TEX>. Similarly, it is universal with respect to differentiation if for any g, R, and <TEX>$\epsilon$</TEX>, there is n such that <TEX>$<TEX>$\mid$</TEX>f^{(n)}(z)\;-\;g(z)<TEX>$\mid$</TEX>\;<\;{\epsilon}\;for\;<TEX>$\mid$</TEX>z<TEX>$\mid$</TEX>\;{\leq}\;R$</TEX>. In this note, we review G. …