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A simple approach to cardinal Lagrange and periodic Lagrange splines. (English) Zbl 0624.41011
The author considers the cardinal spline interpolation problem of finding an element in \(S_ n\) which interpolates to some given function f at the integers Z, where \(s_ n\) is the space of cardinal splines of degree n having (simple) knots at \(-h+Z\), with \(0\leq h<1\). An explicit expression is derived by using a simple approach to the cardinal Lagrange spline as a combination of decreasing null splines and a polynomial correction term near the origin. Many qualitative properties, new and old, are obtained as immediate consequences. The periodic analogue is also discussed. This method is much simpler than the previous work and is readily generalized to cardinal L-splines.
Reviewer: R.S.Dahiya

MSC:
41A15 Spline approximation
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[1] de Boor, C, On the cardinal spline interpolant to eiut, SIAM J. math. anal., 7, 6, 930-941, (1976) · Zbl 0341.41008
[2] de Boor, C; Schoenberg, I.J, Cardinal interpolation and spline functions, VIII. the budan-Fourier theorem for splines and applications, (), 1-17 · Zbl 0319.41010
[3] Meinardus, G; Merz, G, ()
[4] Micchelli, C.A, Oscillation matrices and cardinal spline interpolation, (), 163-201
[5] Michelli, C.A, Cardinal L-splines, (), 203-250
[6] ter Morsche, H, On the relations between finite differences and derivatives of cardinal spline functions, () · Zbl 0315.41009
[7] Nilson, E.N, Polynomial splines and a fundamental eigenvalue problem for polynomials, J. approx. theory, 6, 439-465, (1972) · Zbl 0246.41017
[8] Richards, F, Best bounds for the uniform periodic spline interpolation operator, J. approx. theory, 7, 302-317, (1973) · Zbl 0252.41008
[9] Reimer, M, Extremal spline bases, J. approx. theory, 36, 91-98, (1982) · Zbl 0492.41018
[10] Reimer, M, The radius of convergence of a cardinal Lagrange spline series of odd degree, J. approx. theory, 39, 289-294, (1983) · Zbl 0519.41013
[11] Schoenberg, I.J, Cardinal interpolation and spline functions, IV. the exponential Euler splines, (), 382-404 · Zbl 0268.41004
[12] Schoenberg, I.J, Cardinal interpolation and spline functions, VII. the behavior of spline interpolants as their degree tends to infinity, J. anal. math., 27, 205-229, (1974) · Zbl 0303.41008
[13] Schoenberg, I.J, On Micchelli’s theory of cardinal L-splines, (), 251-276 · Zbl 0343.41009
[14] Siepmann, D; S√ľndermann, B, On a minimal property of cubic periodic Lagrangian splines, J. approx. theory, 39, 236-240, (1983) · Zbl 0521.41007
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