The basic B-spline function.

    Construction of cubic B-spline interpolation can be accomplished by first considering the following basic function.

The function is a piecewise continuous on the interval , it is zero elsewhere.  In advanced courses this simple concept is glamorized by saying that is a function with "compact support."  That is, it is supported (or non-zero) only on a small set.  

The graph of the function .

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Verify that   is a cubic spline.

Each part of   is piecewise cubic.
Are the functions  ,    and    continuous for all   ?
Since    is composed of the piecewise functions ,,,,,, all that is necessary is to see if they join up properly at the nodes  .  However, this will take 15 computations to verify.  This is where Mathematica comes in handy.  Follow the link below if you are interested in the proof.  

The above proof that    is a cubic spline used the formulas ,,,,,.
It is the analytic way to do things and illustrates "precise mathematical" reasoning.

If you trust graphs, then just look at the graphs of  ,  ,  and   and try to determine if they are continuous.  
However, there might be problems lurking about.  If you seek the mathematical truth you should look at the next cell link.  

Let's translate the B-spline over to the node and use the uniform step size .  

Now form the linear combination for the spline.

At each of the nodes for   computation will reveal that

If the B-spline is to go through the points   for  , then the following equations must hold true

       for   .  

For the natural cubic spline, we want the second derivatives to be zero at the left endpoint  .  

Therefore we must have   .  

Computation will reveal that

    

To construct the natural cubic spline, we must have

    .

We can solve this equation for the spline coefficient

For the natural cubic spline, we want the second derivatives to be zero at the right endpoint  .   

Therefore we must have   .  

Computation will reveal that

    

To construct the natural cubic spline, we must have

    .

We can solve this equation for the spline coefficient

The above construction shows how to calculate all the coefficients .

Method I.  B-spline construction using equations.

Illustration using 7 knots.

The following example uses  n = 6.  There are  n+3 = 9  equations to solve and  n+1 = 7  data points or knots.  First set up the 9 equations to be solved.