Shape-preserving interpolation by cubic splines

We consider the problem of shape-preserving interpolation by cubic splines. We propose a unified approach to the derivation of sufficient conditions for the k-monotonicity of splines (the preservation of the sign of any derivative) in interpolation of k-monotone data for k = 0, …, 4.     

Approximation by local <Emphasis Type="Italic">L</Emphasis>-splines corresponding to a linear differential operator of the second order

For the class of functions

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, where \(\mathcal{L}_2 (D)\) is a linear differential operator of the second order whose characteristic polynomial has only real roots, we construct a noninterpolating linear positive method of exponential spline approximation possessing extremal and smoothing properties and locally inheriting the monotonicity of the initial data (the values of a function

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at the points of a uniform grid). The approximation error is calculated exactly for this class of functions in the uniform metric.

     

A Method for the Construction of Wavelet Analogs by Means of Trigonometric <Emphasis Type="Italic">B</Emphasis>-Splines

We construct an analog of two-scale relations for basis trigonometric splines with uniform knots corresponding to a linear differential operator of order 2r + 1 with constant coefficients L_2r+1(D) = D(D₂ + α₁² )(D² + α₂² )... (D² + α _r ² ), where α₁, α₂,..., α _r are arbitrary positive numbers. The properties of nested subspaces of trigonometric splines are analyzed.     

Widths of classes of convolutions with Poisson kernel

Translated from Matematicheskie Zametki, Vol. 51, No. 6, pp. 126–136, June, 1992.     

Lebesgue Constant of Local Cubic Splines with Equally Spaced Nodes

It is proved that the uniform Lebesgue constant (the norm of a linear operator from C to C) of local cubic splines with equally spaced nodes, which preserve cubic polynomials, is equal to 11/9.     

A problem of extremal interpolation

Mathematical Notes -