Functions of variable bandwidth are defined as elements of certain modulation spaces. The latter consist of functions or distributions \(f\) whose short-time Fourier transforms \(V_g f(x,\omega)= \int f(t) \, e^{-2\pi i \omega t}\overline{g (t-x)}\, dt = \langle f, \pi (x,\omega) g\rangle\), using a window \(g\), belong to the weighted mixed-norm space \(M_m^{p,q}(\mathbb{R})\) normed by
\[ \|F\|_{M_m^{p,q}} = \biggl(\biggl( \int_{-\infty}^\infty |F(x,\omega)|^p m(x,\omega)^p \, dx\biggr)^{q/p} \, d\omega\biggr)^{1/q} . \]
Here, \(m\) is assumed to satisfy a certain moderate growth condition that allows one to conclude certain containment properties of the spaces, make conclusions about the equivalence of modulation spaces having corresponding equivalent weights, and allow some flexibility in the choice of the window function \(g\). The exponents \(p,q\) are taken in the range \([1,\infty)\) or, properly modified, \(p\) or \(q=\infty\), to make \(M_m^{p,q}(\mathbb{R})\) a Banach space. This work deals with particular weights of the form \(m_{b,s}(z) = (1+d_b(z))^s\), where \(s>0\) and, for \(z=(x,\omega)\), \(d_b(z)= 0\) if \(|\omega|\leq b(x)\) and \(d_b(z)= |\omega-b(x)|\) if \(|\omega|> b(x)\). Here, \(b(x)\geq 0\) is regarded as a variable bandwidth indicator. The space \(\text{VB}^{p,q}_{m_{b,s}}\) is defined as those \(f\) whose short-time Fourier transforms \(V_g f(x,\omega)\) belong to \(M_m^{p,q}(\mathbb{R})\) with \(m=m_{b,s}\). It is shown that \(\text{VB}^{p,q}_{m_{b,s}}\) increases as the exponents \(p\) or \(q\) increase, decreases as the growth index \(s\) increases, and increases as the bandwidth parameter \(b(x)\) increases pointwise. For the particular case \(p=q=2\), it is shown how to build elements of \(\text{VB}^{2,2}_{m_{b,s}}\) by patching together a sequence of bandlimited functions in \(L^2\) having different bandwidths via a partition of unity. To provide a sense in which the \(\text{VB}\) spaces are a natural way in which to quantify variable bandwidth, approximation error estimations are provided. Given a dense enough time-frequency lattice \(\Lambda\) and appropriate \(g\), the operator \(Sf=\sum_{\lambda\in\Lambda} \langle f,\pi(\lambda) g\rangle \pi (\lambda) g\) is continuously invertible in \(L^2\) and \(\gamma=S^{-1} g\), called the canonical dual of \(g\), satisfies \(f= \sum_{\lambda\in\Lambda} \langle f,\pi(\lambda) \gamma \rangle \pi (\lambda) g\). For the case of \(\text{VB}^{2,2}_{m_{b,s}}\)-spaces, one has that, if \((g,\gamma)\) define a \(\Lambda\)-dual pair of windows in the weighted modulation space \(M_\nu^{1,1}(\mathbb{R})\) and \(0<a<s\), then for any \(\varepsilon>0\) there exists \(r>0\) such that for all \(f\in \text{VB}^{2,2}_{m_{b,s}}\), one has the approximation estimate
\[ \biggl\|f-\sum_{\Lambda\cap ST_{b+r}} \langle f, \, \pi(\lambda)\gamma\rangle \pi(\lambda) g\biggr\|_{\text{VB}^{2,2}_{m_{b,s}}} \leq \varepsilon \|f\|_{\text{VB}^{2,2}_{m_{b,s}}}. \]
Here, \(ST_{b+r}=\{(x,\omega):\, |\omega|\leq b(x)+r \}\) and \(\Lambda\) is a discrete lattice. Corresponding approximation results are also proved for more general \(\text{VB}^{p,q}_{m_{b,s}}\)-spaces.