View source for Bochner almost-periodic functions

Functions equivalent to [[Bohr almost-periodic functions|Bohr almost-periodic functions]]; defined by S. Bochner [[#References|[1]]]. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016700/b0167001.png" /> which is continuous in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016700/b0167002.png" /> is said to be a Bochner almost-periodic function if the family of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016700/b0167003.png" /> is compact in the sense of uniform convergence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016700/b0167004.png" />, i.e. if it is possible to select from each infinite sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016700/b0167005.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016700/b0167006.png" /> a subsequence which converges uniformly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016700/b0167007.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016700/b0167008.png" />. Bochner's definition is extensively employed in the theory of almost-periodic functions; in particular, it serves as the starting point in abstract generalizations of the concept of almost-periodicity.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Bochner,   "Beiträge zur Theorie der fastperiodischen Funktionen I, Funktionen einer Variablen"  ''Math. Ann.'' , '''96'''  (1927)  pp. 119–147</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.M. Levitan,   "Almost-periodic functions" , Moscow  (1953)  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Maak,   "Fastperiodische Funktionen" , Springer  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Amerio,   G. Prouse,   "Almost-periodic functions and functional equations" , v. Nostrand  (1971)</TD></TR></table>