  The main object under consideration is a class $\Phi_n\backslash\Phi_{n+1}$
of radial positive definite functions on $\R^n$ which do not admit \emph{radial
positive definite continuation} on $\R^{n+1}$. We find certain necessary and
sufficient conditions for the Schoenberg representation measure $\nu_n$ of
$f\in \Phi_n$ in order that the inclusion $f\in \Phi_{n+k}$, $k\in\N$, holds.
We show that the class $\Phi_n\backslash\Phi_{n+k}$ is rich enough by giving a
number of examples. In particular, we give a direct proof of
$\Omega_n\in\Phi_n\backslash\Phi_{n+1}$, which avoids Schoenberg's theorem,
$\Omega_n$ is the Schoenberg kernel. We show that
$\Omega_n(a\cdot)\Omega_n(b\cdot)\in\Phi_n\backslash\Phi_{n+1}$, for $a\not=b$.
Moreover, for the square of this function we prove surprisingly much stronger
result: $\Omega_n^2(a\cdot)\in\Phi_{2n-1}\backslash\Phi_{2n}$. We also show
that any $f\in\Phi_n\backslash\Phi_{n+1}$, $n\ge2$, has infinitely many
negative squares. The latter means that for an arbitrary positive integer $N$
there is a finite Schoenberg matrix $\kS_X(f) :=
\|f(|x_i-x_j|_{n+1})\|_{i,j=1}^{m}$, $X := \{x_j\}_{j=1}^m \subset\R^{n+1}$,
which has at least $N$ negative eigenvalues.
