Buescu, JorgePaixão, António2011-11-282011-11-282011-03-01Buescu J, Paixão A C. On differentiability and analyticity of positive definite functions. Journal of Mathemical Analysis and Applications. 2010; 375 (1): 336-341.0022-247Xhttp://hdl.handle.net/10400.21/674We derive a set of differential inequalities for positive definite functions based on previous results derived for positive definite kernels by purely algebraic methods. Our main results show that the global behavior of a smooth positive definite function is, to a large extent, determined solely by the sequence of even-order derivatives at the origin: if a single one of these vanishes then the function is constant; if they are all non-zero and satisfy a natural growth condition, the function is real-analytic and consequently extends holomorphically to a maximal horizontal strip of the complex plane.engPositive definite functionsInequalitiesAnalytic functionsjournal article