The scientific basis is given by applying the concept of visual variables described in the scientific discipline of information visualization.
Indefinite integral of Bernstein basis is given by (2.8).
A more appropriate basis is given by the Prolate Spheroidal Wave functions (PSWFs) [15].
In our example, the maximal incoherence associated to the Fourier basis is given by the canonical or spike basis φ k (t) = δ(t -- k).
Given an N pixel image x, the GBT orthonormal basis is given by the eigenvectors r i,i=1,…,N, diagonalizing the matrix A built as follows: A = diag ∑ j = 1 N l 0, j, ∑ j = 1 N l 1, j, …, ∑ j = 1 N l N, j − L. where L is a binary N×N adjacency matrix whose element lu,v is set to 1 if the pixels u and v are not separated by an image edges, and it is set to 0 otherwise.
The dimension of the polynomial ring F[x] introduced above is countably infinite, a basis is given by,,, A fortiori, the dimension of more general function spaces, such as the space of functions on some (bounded or unbounded) interval, is infinite.
Conditions for constructing a Riesz wavelet basis are given by the following lemma.
The detailed theoretical discussion of thermal expansion on this basis was given by Barron [2].
Note that the weight of each basis function is given by the corresponding coefficient in w i, and the estimated w i is sparse.
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].
Proof The inner product with sinc basis element is given by 〈 λ ( x ) 0 C D x α y ( x ), S k 〉 = ∫ 0 1 ( λ ( x ) S k ( x ) w ( x ) ) 0 C D x α y ( x ) d x.
