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Note that the weight of each basis function is given by the corresponding coefficient in w i, and the estimated w i is sparse.
The quintic B-spline basis function is given in Sect. 2. The finite difference approximation for time discretization of the given problem is discussed in Sect.
Recently, spectral method were developed to solve partial fractional differential equations in which the choice of the basis function is given by means of Jacobi polynomials; see [30, 31].
The two-scale equations for scaling and wavelet basis function are given as follows: (3) φ t = 2 ∑ m ∈ Z h m φ 2 t − m, (4) ψ t = 2 ∑ m ∈ Z g m φ 2 t − m, where g m = (−1) m h1− m.
The procedure for modifying the basis functions is given as follows: B ˜ 0 x = B 0 x + 2 B − 1 x Open image in new window B ˜ 1 x = B 1 x − B − 1 x Open image in new window B ˜ j x = B j x, j = 2, …, N − 2 Open image in new window (3.1) B ˜ N − 1 x = B N − 1 x − B N + 1 x Open image in new window B ˜ N x = B N x + 2 B N + 1 x Open image in new window.
The closed-form expression of the basis functions is given in the Supplementary Information.
Next, we show that the optimal non-overlapping basis functions in the minimal mean square error sense can be found by solving an eigenvalue problem without numerical search when the feasible overlapping basis functions are given.
The schematic presentation of "3D motions" through the space of block-submatrix and locating the assigned HS function on the basis of translators and the starting hyperspherical function is given in Figure 4.
