Properties of the B-spline:

Properties of the B-spline Functions are following

• Local Support Property: N_i, p (u) = 0 for u outside the interval [u_i, u_{i + p+1}). This property might be deduced from the above observation that N_i, p (u) is a linear combination of N_i, _p-1 (u) and Ni + 1, _p-1 (u).
• In any specified knot span, at most p + 1 of the basic functions are non-zero. that basis functions might be non-zero in the knot span [u_j, u_j+1)?
• Non-negativity : N_i, p (u)  for all i, p and u. it is proved easily utilizing induction on p.
• Partition of Unity : For any knot span, [u_i, u_i+1), total of N_j, _p (u) = 1.
• Continuity : All of derivatives of N_i, _p (u) exist in the interior of any knot span (As it is a polynomial).

At a knot, N_i, _p (u) is (p - k) times differentiable, here k is the multiplicity of the knot (i.e., the number of knots of the similar value is the multiplicity of the knot at that value).