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In this work we extend some ideas about greedy algorithms, which are well-established tools for, e.g., kernel bases, and exponential-polynomial splines whose main drawback consists in possible overfitting and consequent oscillations of the approximant. To partially overcome this issue, we develop some results on theoretically optimal interpolation points. Moreover, we introduce two algorithms which perform an adaptive selection of the spline interpolation points based on the minimization either of the sample residuals (f-greedy), or of an upper bound for the approximation error based on the spline Lebesgue function (λ -greedy). Both methods allow us to obtain an adaptive selection of the sampling points, i.e., the spline nodes. While the f-greedy selection is tailored to one specific target function, the λ -greedy algorithm enables us to define target-data-independent interpolation nodes. [ABSTRACT FROM AUTHOR]

In this article, two new techniques of approximation of the reliability of a two-terminal network are developed based on the constructive theory of functions and related methods. Two methods of generating an approximation cubic spline are used: Lagrange-type interpolation procedures and Bernstein approximation operator. A possibility of minimizing the total error of approximation, based on keeping some properties invariant, is described in case of a large class of pairs of dual two-terminal networks. Simulations are included, showing that the error of approximation is negligible in case of some special initial data. [ABSTRACT FROM AUTHOR]

Time delay or the time-of-flight is a most frequently used parameter in many ultrasonic applications. Delay estimation is based on the sampled signal, so the resolution is limited by the sampling grid. Higher accuracy is available if the signal-to-noise ratio is high, then the subsample estimate is desired. Techniques used for subsample interpolation suffer from bias error. Time-of-flight estimation that is free from bias errors is required. The proposed subsample estimation works in the frequency domain; it is based on the cross-correlation peak temporal position. The phase of the cross-correlation frequency response becomes linear thanks to multiplication by the complex conjugate, and its inclination angle is proportional to the delay. Then, subsample interpolation becomes free from the bias error. Twelve algorithmic implementations of this technique have been proposed in this paper. All algorithmic implementations have been analyzed for bias and random errors using simulation and MATLAB codes are given as supplementary material. Comparison with best-performing interpolation techniques (spline approximation, cosine interpolation, carrier phase) is given for both bias and random errors. It was demonstrated that frequency domain interpolation has no bias errors, and noise performance is better or comparable to other subsample estimation techniques. Weighted regression using L2 norm minimization has the best performance: total errors (bias and random) are within 3% of theoretical lower bound. [ABSTRACT FROM AUTHOR]

To reduce the feedrate fluctuation in non-uniform rational B-spline (NURBS) interpolation caused by approximation errors of interpolation methods, a new NURBS interpolation method with minimal feedrate fluctuation for CNC machine tools is proposed to obtain the sampling curve parameter by solving a quartic equation with respect to the curve parameter increment according to the scheduled feedrate using Ferrari's formulas and Shengjin's formulas. Firstly, the second-order Taylor expansion is applied on the numerator and denominator in the NURBS representation separately with unknown curve parameter increment. Then, a quartic equation with respect to the curve parameter increment is formed and solved by Ferrari's formulas and Shengjin's formulas efficiently, which can achieve minimal feedrate fluctuation in real-time interpolation. In principle, the proposed method can totally eliminate the feedrate fluctuation for second-degree NURBS curves and can reduce it to a satisfied value for other high-degree NURBS curves. Finally, numerical simulations and experiments are conducted to verify the feasibility and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the approximation of a function by its interpolating multilinear spline and the approximation of its derivatives by the derivatives of the corresponding spline. We obtain the exact uniform approximation error on classes of functions with moduli of continuity bounded above by certain majorants. [Copyright &y& Elsevier]

In this paper, we formally investigate two mathematical aspects of Hermite splines that are relevant to practical applications. We first demonstrate that Hermite splines are maximally localized, in the sense that the size of their support is minimal among pairs of functions with identical reproduction properties. Then, we precisely quantify the approximation power of Hermite splines for the reconstruction of functions and their derivatives. It is known that the Hermite and B-spline approximation schemes have the same approximation order. More precisely, their approximation error vanishes as O (T 4) when the step size T goes to zero. In this work, we show that they actually have the same asymptotic approximation error constants, too. Therefore, they have identical asymptotic approximation properties. Hermite splines combine optimal localization and excellent approximation power, while retaining interpolation properties and closed-form expression, in contrast to existing similar functions. These findings shed a new light on the convenience of Hermite splines in the context of computer graphics and geometrical design. [ABSTRACT FROM AUTHOR]