Let
$\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\in \mathbb{R}$
and
$s\in \mathbb{N}$
be given. Let
$\unicode[STIX]{x1D6FF}_{x}$
denote the Dirac measure at
$x\in \mathbb{R}$
, and let
$\ast$
denote convolution. If
$\unicode[STIX]{x1D707}$
is a measure,
$\unicode[STIX]{x1D707}^{\star }$
is the measure that assigns to each Borel set
$A$
the value
$\overline{\unicode[STIX]{x1D707}(-A)}$
. If
$u\in \mathbb{R}$
, we put
$\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},u}=e^{iu(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FD})/2}\unicode[STIX]{x1D6FF}_{0}-e^{iu(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})/2}\unicode[STIX]{x1D6FF}_{u}$
. Then we call a function
$g\in L^{2}(\mathbb{R})$
a generalized
$(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$
-difference of order
$2s$
if for some
$u\in \mathbb{R}$
and
$h\in L^{2}(\mathbb{R})$
we have
$g=[\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},u}+\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},u}^{\star }]^{s}\ast h$
. We denote by
${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$
the vector space of all functions
$f$
in
$L^{2}(\mathbb{R})$
such that
$f$
is a finite sum of generalized
$(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$
-differences of order
$2s$
. It is shown that every function in
${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$
is a sum of
$4s+1$
generalized
$(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$
-differences of order
$2s$
. Letting
$\widehat{f}$
denote the Fourier transform of a function
$f\in L^{2}(\mathbb{R})$
, it is shown that
$f\in {\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$
if and only if
$\widehat{f}$
“vanishes” near
$\unicode[STIX]{x1D6FC}$
and
$\unicode[STIX]{x1D6FD}$
at a rate comparable with
$(x-\unicode[STIX]{x1D6FC})^{2s}(x-\unicode[STIX]{x1D6FD})^{2s}$
. In fact,
${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$
is a Hilbert space where the inner product of functions
$f$
and
$g$
is
$\int _{-\infty }^{\infty }(1+(x-\unicode[STIX]{x1D6FC})^{-2s}(x-\unicode[STIX]{x1D6FD})^{-2s})\widehat{f}(x)\overline{\widehat{g}(x)}\,dx$
. Letting
$D$
denote differentiation, and letting
$I$
denote the identity operator, the operator
$(D^{2}-i(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})D-\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}I)^{s}$
is bounded with multiplier
$(-1)^{s}(x-\unicode[STIX]{x1D6FC})^{s}(x-\unicode[STIX]{x1D6FD})^{s}$
, and the Sobolev subspace of
$L^{2}(\mathbb{R})$
of order
$2s$
can be given a norm equivalent to the usual one so that
$(D^{2}-i(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})D-\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}I)^{s}$
becomes an isometry onto the Hilbert space
${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$
. So a space
${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$
may be regarded as a type of Sobolev space having a negative index.