A family of nearest-neighbor interpolations

Let an integer k range along a survey line, and let data values x_k be packed into a vector $\bold x$.(Each data point x_k could also be a seismogram.) We plan to resample the data more densely, say from 4 to 6 points. For illustration, I follow a crude nearest-neighbor interpolation

scheme by sprinkling ones along the diagonal of a rectangular matrix that is  

$$\bold y \eq \bold B \, \bold x$$ (10)

where  

$$\left[ \begin{array} {c} y_1 \ y_2 \ y_3 \ y_4 \... ...{array} {c} x_1 \ x_2 \ x_3 \ x_4 \end{array} \right]$$ (11)

The interpolated data is simply $\bold y = (x_1, x_2,x_2,x_3, x_4,x_4)$.The matrix multiplication (11) would not be done in practice. Instead there would be a loop running over the space of the outputs $\bold y$ that picked up values from the input.