Vanishing moments of wavelet packets and wavelets associated with Riesz projectors

Fourier–Boas-Like Wavelets and Their Vanishing Moments

“…New wavelet functions, called Fourier-Boas-Like wavelets, have been established. ese wavelets have been found to be better than the earlier proposed wavelets in [12,21] derived from the Riesz projectors and Fourier kernels, respectively. Various results related to higher vanishing moments of Fourier-Boas-Like wavelets have been given, and it is observed that regularity and fast decay are significant attributes for the vanishing moments of Fourier-Boas-Like wavelets.…”

“…Further, a factor of (1/ � 2 √ ) was imposed on function ψ(x) − iHψ(x) in order to aver the same energy and admissibility coefficient of its generating wavelet. Later, Khanna et al [12] defined an improved and natural version of such wavelets by employing Riesz projectors on wavelets. e main idea behind these wavelets was to perlustrate both even and odd symmetries of an asymmetric signal.…”

“…It is well established in [5] that a pair of quadrature mirror filter coefficients, (a k ) k∈Z , (b k ) k∈Z ∈ l 2 (Z), is associated to the MRA and the following relations ϕ(η) � m 0 (η/2)ϕ(η/2), ψ(η) � m 1 (η/2)ϕ(η/2) are satisfied for η ∈ R, where m 0 and m 1 are given by m 0 (η) � k∈Z a k e ikη , m 1 (η) � k∈Z b k e ikη � e iη m 0 (η + π). For more details on wavelets, one may refer to [6][7][8][9][10][11][12][13][14].…”

“…Recall from [12] that wavelets associated with projection operators P + and P − involving the Hilbert transform are defined as P + � (1/2)(I + iH) and P − � (1/2)(I − iH), where I and H represent the identity operator and Hilbert transform operator, respectively. ese projection operators are also known as Riesz projectors.…”

“…Note that φ is said to be a convolution filter if φ(x) � w * x for any x, where w ∈ (L 1 ∩ L 2 )(R) is a weight function and w(η) is known as the transfer function. For more details, one may see [12].…”