def parabolic(sample_array, peak_index):
    """
    Quadratic interpolation for estimating the true position of an
    inter-sample local maximum when nearby samples are known.
   
    Parameters
    ----------
    sample_array : array_like (1D)
        Array of samples.
    peak_index : int
        Index for the local maximum in sample_array for which to estimate the inter-sample maximum.
   
    Returns
    -------
    tuple
        The (x,y) coordinates of the vertex of a parabola through peak_index and its two neighbors.
    """
    vertex_x = 1/2. * (sample_array[peak_index-1] - sample_array[peak_index+1]) / (sample_array[peak_index-1] - 2 * sample_array[peak_index] + sample_array[peak_index+1]) + peak_index
    vertex_y = sample_array[peak_index] - 1/4. * (sample_array[peak_index-1] - sample_array[peak_index+1]) * (vertex_x - peak_index)
    return (vertex_x, vertex_y)