R: Object of class 'fts.timedom'

fts.timedom {freqdom.fda}

R Documentation

Object of class `fts.timedom`

Description

Creates an object of class fts.timedom. This object corresponds to a sequence of linear operators.

Usage

fts.timedom(A, basisX, basisY = basisX)

Arguments

`A`	an object of class timedom.
`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 functional linear filter, i.e. a sequence of linear operators, each of which corresponds to a certain lag. Formally we consider an object of class timedom and add some basis functions. Depending on the context, we have different interpretations for the new object.

(I) In order to define operators which maps between two functions spaces, the we interpret A$operators as coefficients in the basis function expansion of the kernel of some finite rank operators

\mathcal{A}_k:\mathrm{span}(\code{basisY})\to\mathrm{span}(\code{basisX}).

The kernels are a_k(u,v)=\boldsymbol{b}_1^\prime(u)\, A_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 lags \ell_1<\ell_2<\cdots<\ell_K. The object this function creates corresponds to the mapping \ell_k \mapsto a_k(u,v).

(II) We may ignore basisX, and represent the linear mapping

\mathcal{A}_k:\mathrm{span}(\code{basisY})\to R^{d_1},

by considering a_k(v):=A_k\,\boldsymbol{b}_2(v) and \mathcal{A}_k(x)=\int a_k(v)x(v)dv.

(III) We may ignore basisY, and represent the linear mapping

\mathcal{A}_k: R^{d_1}\to\mathrm{span}(\code{basisX}),

by considering a_k(u):=\boldsymbol{b}_1^\prime(u)A_k and \mathcal{A}_k(y)=a_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 A$operators.
basisX \quad returns basisX as given in the argument.
basisY \quad returns basisY as given in the argument.
lags \quad returns A$lags.