... Coifman scalet ϕ and wavelet ψ(L = 4).
MEYER WAVELETS 75
3.9 MEYER WAVELETS
In the previous sections, the Battle–Lemarie, Daubechies, and Coifman wavelets
were derived and expressed in the time ... for wavelets, but it is stated here for ease of refer-
ence. These properties will be used in later sections of this chapter to construct the
Daubechies and Coifman wavelets.
3.6 DAUBECHIE...
... W
s,2
()-Sobolev space equipped with inner product of
u,v
s,2
:=
|α |≤s
D
α
uD
α
vd
1
Wavelets in Electromagnetics and Device Modeling. George W. Pan
Copyright
¶ 2003 John Wiley & Sons, ... important too, since it ensures the exis-
tence of the inverse.
1.2.8 Linear Operators
In computational electromagnetics, the method of moments and finite element
method are based on line...
... surfaces,” J. Opt. Soc. Am., 2(12), 2274–2284, Dec. 1985.
[27] G. Pan, “Orthogonal wavelets with applications in electromagnetics, ” IEEE Trans.
Magn., 32, 975–983, 1996.
[28] E. Thorsos, “Backscattering ... the
Galerkin procedure. More specifically, we choose the Coifman wavelets, which are
orthogonal and compactly supported with zero moments of both the wavelets and
scalets. As a conse...
... sets [20], and wavelets on curved surfaces as in the case of spheri-
cal wavelets [21].
2.3 THE HAAR WAVELETS AND MULTIRESOLUTION ANALYSIS
One of the most important properties of wavelets is the ... book, we will refer to
mother wavelets as wavelets and scaling functions as scalets, in order to emphasize
their roles as counterparts of wavelets. Notice that the term wavelets has a...
... out using standard
wavelets. As long as the boundary curve has a closed contour, there is no need to
employ the intervallic wavelets, nor the periodic wavelets. The standard wavelets
are sufficient. ... the wavelets.
In this section we will discuss the FWT using the Franklin (or more generally the
Battle–Lemarie) wavelets, although the technique is applicable to other wavelets.
The F...
... of the finite difference concept.
5.4 FDTD WITH BATTLE–LEMARIE WAVELETS
Battle–Lemarie wavelets possess better regularity than Haar wavelets. The Battle–
Lemarie based time domain method, referred ... (based on CFD), MRTD-9, MRTD-16 (based
on Battle–Lemarie wavelets of N = 9andN = 16 in (5.8.3)), SBTD and CDF2-2
(based on CDF2-2 biorthogonal wavelets [16]). The results are listed in T...
... possessed by any wavelets, as proved
in [1].
To overcome the limitations of the regular (i.e., scalar) wavelets, mathematicians
have proposed multiwavelets. There are two categories of multiwavelets, ... VECTOR-MATRIX DILATION EQUATION
Multiwavelets offer more flexibility than traditional wavelets by extending the scalar
dilation equation
ϕ(t) =
h
k
ϕ(2t − k)
240
Wavelets in Electroma...
... structures with
geometric constraints, modified wavelets, including periodic wavelets and interval-
lic wavelets, were introduced [8, 27]. Nevertheless, the modified wavelets, or the
wavelet-like bases [28] ... standard wavelets, includ-
ing orthogonality, vanishing moments, MRA, single-point quadrature, and the like,
are preserved. The localized correction current is numerically evaluate...
... subsection. When using wavelets on the real line to solve problems
with finite intervals, improper selection of the wavelets can result in nonphysical so-
lutions. In contrast, any type of wavelets on the ... discontinuity space for that
contour. We must employ intervallic wavelets, instead of periodic wavelets.
Since the periodized wavelets are defined in L
2
([0, 1]), one must map...
... it is again interpolated from the coarser
scale in the SPR.
10.4 INTERPOLATION WAVELETS IN THE FDM
Interpolation wavelets can be applied to the finite difference methods (FDM), and
differentiation ... The 1D problem has been discretized
using the SPR with interpolating wavelets. Unlike analytic solutions, in which as-
488 WAVELETS IN NONLINEAR SEMICONDUCTOR DEVICES
flowing from node (i,...