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- W23104550 abstract "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. MacLane's proof of the existence of universal functions with respect to differentiation, and we give a simplified proof of G. D. Birkhoff's theorem showing the existence of universal functions with respect to translation. We also discuss Godefroy and Shapiro's extension of these results to convolution operators as well as some new, related results and problems." @default.
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- W23104550 date "2004-01-01" @default.
- W23104550 modified "2023-10-15" @default.
- W23104550 title "ON UNIVERSAL FUNCTIONS" @default.
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- W23104550 doi "https://doi.org/10.4134/jkms.2004.41.1.065" @default.
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