R: Creates an object of class 'fts.freqdom'.

fts.freqdom {freqdom.fda}

R Documentation

Creates an object of class `fts.freqdom`.

Description

Creates an object of class fts.freqdom.

Usage

fts.freqdom(F, basisX, basisY = basisX)

Arguments

`F`	an object of class freqdom.
`basisX`	an object of class `basis.fd` (see `create.basis`)
`basisY`	an object of class `basis.fd` (see `create.basis`)

Details

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),\ldots, b_{1d_1}(u))^\prime and \boldsymbol{b_2}(u)=(b_{21}(u),\ldots, b_{2d_1}(u))^\prime are the basis functions provided by the arguments basisX and basisY, respectively. Moreover, we consider frequencies \{\omega_1,\ldots, \omega_K\}\subset[-\pi,\pi]. The object this function creates corresponds to the mapping \omega_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.

Value

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.