Using the wavelet transform defined in the infinite domain to process the signal defined in finite interval, the wavelet transform coefficients at the boundary are usually very large. It will bring severe boundary effect, which reduces the calculation accuracy. The construction of interval wavelet is the most common method to reduce the boundary effect. By studying the properties of Shannon-Cosine interpolation wavelet, an improved version of the wavelet function is proposed, and the corresponding interval interpolation wavelet based on Hermite interpolation extension and variational principle is designed, which possesses almost all of the excellent properties such as interpolation, smoothness, compact support and normalization. Then, the multi-scale interpolation operator is constructed, which can be applied to select the sparse feature points and reconstruct signal based on these sparse points adaptively. To validate the effectiveness of the proposed method, we compare the proposed method with Shannon-Cosine interpolation wavelet method, Akima method, Bezier method and cubic spline method by taking infinitesimal derivable function $cos\left(x\right)$ and irregular piecewise function as an example. In the reconstruction of $cos\left(x\right)$ and piecewise function, the proposed method reduces the boundary effect at the endpoints. When the interpolation points are the same, the maximum error, average absolute error, mean square error and running time are $1.20$ × 10${}^{-4}$, $2.52$ × 10${}^{-3}$, $2.76$ × 10${}^{-5}$, $1.68$ × 10${}^{-2}$ and $4.02$ × 10${}^{-3}$, $4.94$ × 10${}^{-4}$, $1.11$ × 10${}^{-3}$, $9.27$ × 10${}^{-3}$, respectively. The four indicators mentioned above are all lower than the other three methods. When reconstructing an infinitely derivable function, the curve reconstructed by our method is smoother, and it satisfies $C^2$ and $G^2$ continuity. Therefore, the proposed method can better realize the reconstruction of smooth curves, improve the reconstruction efficiency and provide new ideas to the curve reconstruction method.

In the case of strong nonlinearity and limited measurement update frequency, the precision of the UKF declines significantly because of decreases in state transition accuracy and correction ability. A smaller step size can improve estimate accuracy, but it will increase the computational load. This paper proposes an adaptive step size UKF for a nonlinear model with continuous time dynamics and discrete time measurements, which can balance precision and computation cost. A novel dynamic nonlinearity detection algorithm is proposed, based on the cosine similarity and the Euclidean distance ratio similarity of a sigma point pair after nonlinear transformation. Meanwhile, in order to avoid the increase of computation due to the nonlinearity of the non-filter performance-critical states, a sigma point construction scheme that considers filter performance-critical states has been designed. Moreover, we introduce a guideline to adaptively select appropriate step sizes based on the degree of nonlinearity in performance-critical states. In the strong nonlinear interval, the method switches to smaller step sizes and takes the obtained prediction into the update at the entire time step. Simulation results demonstrate that under highly nonlinear system conditions, and when the measurement frequency is limited, the proposed method outperforms several reference algorithms in precision and real-time performance.