This paper proposes a construction of local C r interpolation spaces and C r conforming finite element spaces with arbitrary r in any dimension. It is shown that if k ≥ 2 d r +1 the space P k of polynomials of degree ≤ k can be taken as the shape function space of C r finite element spaces in d dimensions. This is the first work on constructing such C r conforming finite elements in any dimension in a unified way. It solves a long-standing open problem in finite element methods.

Abstract-Utilization of window functions and interpolation algorithms for Fast Fourier transform can effectively restrain the spectral leakage and picket fence effect in situation of nonsynchronized sampling and non-interger cycle truncation. In this paper, an improved FFT approach for harmonic based on Nuttall self-convolution window triple-spectral-line interpolation is improved. The simulation results showed that the proposed method have a higher accuracy of harmonic analysis and could improve the anti-noise performance. In the paper ， triplespectral-line interpolation correction algorithm for Nuttall selfconvolution window is used to improve the accuracy and anti noise is the latest research results, by the simulation can prove the superiority of this algorithm.