For instance, if en are any orthonormal basis functions of L2[0,1], then a given function in L2[0,1] can be approximated as a finite linear combination :f(x) \approx f_n (x) = a_1 e_1 (x) + a_2 e_2(x) + \cdots + a_n e_n (x).
Example 2.1 Consider the neural field equation on a d-dimensional cube B = [ 0, 2 π ] d with noise based on trigonometric basis functions of L 2 ( [ 0, 2 π ] d ).
Since the Bernstein basis functions of degree (n+1) are linearly independent, we obtain (beta_{i} = 0) ((i = 0,1, ldots,n + 1)).
The geometry for this example is represented by NURBS basis functions of order 1 in the radial direction and of order 2 in the angular direction, see [20].
Moreover, we note that (u_{2}) in (7) is the 'outer' product of the stimuli (u_{1}), that is, mathbf{D} = mathbf{cc}^{H}, (27) where mathbf{c}= [ c_{-L_{t}}, ldots, c_{ L_{t}} ] ^{T} (28) are the coefficients of the basis functions of (u_{1}).
The key technique is to relate spatial fuzzy basis functions of a 3-D FLC to kernel functions of a SVR and construct an equivalence relationship of a 3-D FLC and a SVR.
The spatially coherent transient deformation field is expanded in Deslauriers-Dubuc (DD) interpolating wavelet basis functions of degree 3 as in Ohtani et al. (2010); only the horizontal components of the GPS time series are analyzed because of significantly larger colored noise in the vertical.
By using basis functions of degree 3 and due to the simple geometry mappings, the exact solution u is in V h.
Therefore, the dimension of (mathcal {P}_{12setminus 11}), the preimage of (mathcal {R}_{12setminus 11}) under H12″, is indeed large enough to allow T 2 to transmit the remaining (d_{2_{mathsf {Int}}}') symbols along the basis functions of (text {dim}mathcal {P}_{12setminus 11}).
Twelve timbre audio features (×1 to ×12) are extracted using the 12 basis functions of timbre vector (Fig. 3) for each segment of an IVS.
