This article presents sufficient conditions for the positive definiteness of radial functions
f
(
x
)
=
φ
(
‖
x
‖
)
f(x) = \varphi (\|x\|)
,
x
∈
R
n
x \in \mathbb {R}^n
, in terms of the derivatives of
φ
\varphi
. The criterion extends and unifies the previous analogues of Pólya’s theorem and applies to arbitrarily smooth functions. In particular, it provides upper bounds on the Kuttner-Golubov function
k
n
(
λ
)
k_n(\lambda )
which gives the minimal value of
κ
\kappa
such that the truncated power function
(
1
−
‖
x
‖
λ
)
+
κ
(1-\|x\|^\lambda )_+^\kappa
,
x
∈
R
n
x \in \mathbb {R}^n
, is positive definite. Analogous problems and criteria of Pólya type for
‖
⋅
‖
α
\|\cdot \|_\alpha
-dependent functions,
α
>
0
\alpha > 0
, are also considered.