Discrete spline interpolation

Let x₁, x₂, . . ., x_n-1 be an increasing sequence of real numbers. Let g(x) be a piecewise polynomial defined by

:$$

g(x)=

\begin{cases}

g_1(x) & x

g_i(x) & x_{i-1}\le x < x_i \text{ for } i = 2,3, \ldots, n-1\\

g_n(x) & x\ge x_{n-1}

\end{cases}

where g₁(x), . . ., g_n(x) are polynomials of degree 3. Let h > 0. If

:$$

(g_{i+1}-g_i)(x_i +jh)=0 \text{ for } j=-1,0,1 \text{ and } i=1,2,\ldots, n-1

then g(x) is called a discrete cubic spline.

The conditions defining a discrete cubic spline are equivalent to the following:

:$g\_\{i+1\}(x\_i-h)\; =\; g\_i(x\_i-h)$

:$g\_\{i+1\}(x\_i)\; =\; g\_i(x\_i)$

:$g\_\{i+1\}(x\_i+h)\; =\; g\_i(x\_i+h)$

The central differences of orders 0, 1, and 2 of a function f(x) are defined as follows:

:$D^\{(0)\}f(x)\; =\; f(x)$

:$D^\{(1)\}f(x)=\backslash frac\{f(x+h)-f(x-h)\}\{2h\}$

:$D^\{(2)\}f(x)=\backslash frac\{f(x+h)-2f(x)+f(x-h)\}\{h^2\}$

The conditions defining a discrete cubic spline are also equivalent to

:$D^\{(j)\}g\_\{i+1\}(x\_i)=D^\{(j)\}g\_i(x\_i)\; \backslash text\{\; for\; \}\; j=0,1,2\; \backslash text\{\; and\; \}\; i=1,2,\; \backslash ldots,\; n-1.$

This states that the central differences $D^\{(j)\}g(x)$ are continuous at x_i.

Let x₁ = 1 and x₂ = 2 so that n = 3. The following function defines a discrete cubic spline:

g(x) =

\begin{cases}

x^3 & x<1 \\

x^3 - 2(x-1)((x-1)^2-h^2) & 1\le x < 2\\

x^3 - 2(x-1)((x-1)^2-h^2)+(x-2)((x-2)^2-h^2) & x \ge 2

\end{cases}

Let x₀ < x₁ and x_n > x_n-1 and f(x) be a function defined in the closed interval [x₀ - h, x_n + h]. Then there is a unique cubic discrete spline g(x) satisfying the following conditions:

:$g(x\_i)\; =\; f(x\_i)\; \backslash text\{\; for\; \}\; i=0,1,\backslash ldots,\; n.$

:$D^\{(1)\}g\_1(x\_0)\; =\; D^\{(1)\}f(x\_0).$

:$D^\{(1)\}g\_n(x\_n)\; =\; D^\{(1)\}f(x\_n).$

This unique discrete cubic spline is the discrete spline interpolant to f(x) in the interval [x₀ - h, x_n + h]. This interpolant agrees with the values of f(x) at x₀, x₁, . . ., x_n.

• Discrete cubic splines were originally introduced as solutions of certain minimization problems.
• They have applications in computing nonlinear splines.{{cite journal|last1=Michael A. Malcolm|title=On the computation of nonlinear spline functions|journal=SIAM Journal on Numerical Analysis|date=April 1977|volume=14|issue=2|pages=254–282|doi=10.1137/0714017|bibcode=1977SJNA...14..254M }}
• They are used to obtain approximate solution of a second order boundary value problem.{{cite journal|last1=Fengmin Chen, Wong, P.J.Y.|title=Solving second order boundary value problems by discrete cubic splines|journal=Control Automation Robotics & Vision (ICARCV), 2012 12th International Conference|date=Dec 2012|pages=1800–1805}}
• Discrete interpolatory splines have been used to construct biorthogonal wavelets.{{cite journal|last1=Averbuch, A.Z., Pevnyi, A.B., Zheludev, V.A.|title=Biorthogonal Butterworth wavelets derived from discrete interpolatory splines|journal=IEEE Transactions on Signal Processing|date=Nov 2001|volume=49|issue=11|pages=2682–2692|doi=10.1109/78.960415|bibcode=2001ITSP...49.2682A |citeseerx=10.1.1.332.7428}}

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Category:Interpolation

Category:Splines (mathematics)