Optimization of a Polynomial Seventh-Order Two-Parameter Interpolation Kernel Using the Extended Continuity Criterion

polynomial seventh-order two-parameter (2P) interpolation kernel provides increased flexibility for approximating the ideal sinc interpolation kernel compared with the corresponding one-parameter (1P) kernel. This paper presents an optimization approach for the seventh-order polynomial 2P interpolation kernel based on the Continuity criterion. The criterion requires continuity of the sixth-order derivative of the seventh-order polynomial kernel at the interpolation node 𝒔=𝟏. Since this condition provides only one equation for the two independent parameters,𝜶 and 𝜷, the Continuity criterion is extended by imposing the same condition at the interpolation node 𝒔=𝟐. The resulting Extended continuity criterion provides a system of two equations whose solution uniquely determines the kernel parameters𝝃 and 𝝃. The performance of the optimized 2P kernel is evaluated in the spectral domain using similarity measures based on the amplitude characteristic, including the total squared error, the slope in the transition band, and ripple measures. The results are compared with those obtained for seventh-order polynomial 1P and 2P kernels optimized according to the Slope and Extended slope criteria, respectively, as well as with the ideal sinc and windowed sinc kernels. The results show that the Extended continuity criterion provides lower RMS ripple and ripple energy, whereas the Extended slope criterion provides lower total squared error and a steeper transition characteristic. The findings demonstrate that both optimization approaches can be effectively applied to the seventh-order polynomial 2P interpolation kernel, with the choice of criterion depending on the required interpolation characteristics.