  A new class of lattice Dirac operators $D$ which satisfy the index theorem
have been recently proposed on the basis of the algebraic relation
$\gamma_{5}(\gamma_{5}D) + (\gamma_{5}D)\gamma_{5} =
2a^{2k+1}(\gamma_{5}D)^{2k+2}$. Here $k$ stands for a non-negative integer and
$k=0$ corresponds to the ordinary Ginsparg-Wilson relation. We analyze the
locality properties of Dirac operators which solve the above algebraic
relation.
  We first show that the free fermion operator is analytic in the entire
Brillouin zone for a suitable choice of parameters $m_{0}$ and $r$, and there
exists a well-defined ``mass gap'' in momentum space, which in turn leads to
the exponential decay of the operator in coordinate space for any finite $k$.
This mass gap in the free fermion operator suggests that the operator is local
for sufficiently weak background gauge fields. We in fact establish a finite
locality domain of gauge field strength for
$\Gamma_{5}=\gamma_{5}-(a\gamma_{5}D)^{2k+1}$ for any finite $k$, which is
sufficient for the cohomological analyses of chiral gauge theory. We also
present a crude estimate of the localization length defined by an exponential
decay of the Dirac operator, which turns out to be much shorter than the one
given by the general Legendre expansion.
