In the mathematical field of numerical analysis, discrete spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a discrete spline. A discrete spline is a piecewise polynomial such that its central differences are continuous at the knots whereas a spline is a piecewise polynomial such that its derivatives are continuous at the knots. Discrete cubic splines are discrete splines where the central differences of orders 0, 1, and 2 are required to be continuous.
Discrete splines were introduced by Mangasarin and Schumaker in 1971 as solutions of certain minimization problems involving differences.
Discrete cubic splines.
Let "x"1, "x"2, . . ., "x""n"-1 be an increasing sequence of real numbers. Let "g"("x") be a piecewise polynomial defined by
formula_1
where "g"1("x"), . . ., "g""n"("x") are polynomials of degree 3. Let "h" ; 0. If
formula_2
then "g"("x") is called a discrete cubic spline.
Alternative formulation 1.
The conditions defining a discrete cubic spline are equivalent to the following:
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Alternative formulation 2.
The central differences of orders 0, 1, and 2 of a function "f"("x") are defined as follows:
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formula_7
formula_8
The conditions defining a discrete cubic spline are also equivalent to
formula_9
This states that the central differences formula_10 are continuous at "x""i".
Example.
Let "x"1 = 1 and "x"2 = 2 so that "n" = 3. The following function defines a discrete cubic spline:
formula_11
Discrete cubic spline interpolant.
Let "x"0 ; "x"1 and "x""n" ; "x""n"-1 and "f"("x") be a function defined in the closed interval ["x"0 - h, "x""n" + h]. Then there is a unique cubic discrete spline "g"("x") satisfying the following conditions:
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This unique discrete cubic spline is the discrete spline interpolant to "f"("x") in the interval ["x"0 - h, "x""n" + h]. This interpolant agrees with the values of "f"("x") at "x"0, "x"1, . . ., "x"n.
References.
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