Summary: While it is well known that the \(m\)-th order \(B\)-spline \(N_ m(x)\) with integer knots generates a multiresolution analysis, \(\cdots\subset V_{- 1}\subset V_ 0\subset\cdots\), with the \(m\)th order of approximation, we prove that \(\psi(x):=L^{(m)}_{2m}(2x-1)\), where \(L_{2m}(x)\) denotes the \((2m)\)th order fundamental cardinal interpolatory spline, generates the orthogonal complementary wavelet spaces \(W_ k\). Note that for \(m=1\), when the \(B\)-spline \(N_ 1(x)\) is the characteristic function of the unit interval \([0,1)\), our basic wavelet \(L_ 2'(2x-1)\) is simply the well-known Haar wavelet. In proving that \(V_{k+1}=V_ k\oplus W_ k\), we give the exact formulation of \(N_ m(2x-j)\), \(j\in\mathbb{Z}\), in terms of integer translates of \(N_ m(x)\) and \(\psi(x)\). This allows us to derive a wavelet decomposition algorithm without relying on orthogonality nor construction of a dual basis.