fts.freqdom | R Documentation |
fts.freqdom
.Creates an object of class fts.freqdom
.
fts.freqdom(F, basisX, basisY = basisX)
F |
an object of class freqdom. |
basisX |
an object of class |
basisY |
an object of class |
This class is used to describe a frequency domain operator (for example a spectral
density operator) on selected frequencies. Formally we consider an object of class
freqdom
and add some basis functions. Depending on the context, we
have different interpretations for the new object.
(I) In order to define an operator which maps between two functions spaces, the we
interpret F$operators
as coefficients in the basis function expansion of
the kernel of some finite rank operators
\mathcal{F}_k:\mathrm{span}(\code{basisY})+\mathrm{i}\, \mathrm{span}(\code{basisY})\to\mathrm{span}(\code{basisX})+\mathrm{i}\, \mathrm{span}(\code{basisX}).
The kernels are f_k(u,v)=\boldsymbol{b}_1^\prime(u)\, F_k\, \boldsymbol{b}_2(v), where \boldsymbol{b_1}(u)=(b_{11}(u),…, b_{1d_1}(u))^\prime and \boldsymbol{b_2}(u)=(b_{21}(u),…, b_{2d_1}(u))^\prime are the basis functions provided by the arguments basisX and basisY, respectively. Moreover, we consider frequencies \{ω_1,…, ω_K\}\subset[-π,π]. The object this function creates corresponds to the mapping ω_k \mapsto f_k(u,v).
(II) We may ignore basisX, and represent the linear mapping
\mathcal{F}_k:\mathrm{span}(\code{basisY})+\mathrm{i}\, \mathrm{span}(\code{basisY})\to C^{d_1},
by considering f_k(v):=F_k\,\boldsymbol{b}_2(v) and \mathcal{F}_k(x)=\int f_k(v)x(v)dv.
(III) We may ignore basisY, and represent the linear mapping
\mathcal{F}_k: C^{d_1}\to\mathrm{span}(\code{basisX})+\mathrm{i}\, \mathrm{span}(\code{basisX}),
by considering f_k(u):=\boldsymbol{b}_1^\prime(u)F_k and \mathcal{F}_k(y)=f_k(u)y.
Returns an object of class fts.freqdom
. An object of class
fts.freqdom
is a list containing the following components:
operators
\quad returns the array F$operators
.
basisX
\quad returns basisX
as given in the argument.
basisY
\quad returns basisY
as given in the argument.
freq
\quad returns the vector F$freq
.
The multivariate equivalent in the freqdom
package: freqdom
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