# multiProj: Meyer wavelet projection given a set of wavelet coefficients In mwaved: Multichannel Wavelet Deconvolution with Additive Long Memory Noise

## Description

Reconstructs a function using wavelet coefficients (`waveletCoef` object) as input.

## Usage

 `1` ```multiProj(beta, j1 = log2(length(beta\$coef)) - 1) ```

## Arguments

 `beta` A `waveletCoef` object that contains a vector of wavelet coefficients and the coarsest resolution level, `j0` to create the required output function expansion. `j1` The finest resolution to be used in the projection (specifies which resolution that the wavelet expansion is truncated).

## Details

Function that takes an input of wavelet coefficients in the form of a `waveletCoef` object (see `multiCoef` for details) and optionally a desired maximum resolution level, `j1`, to create an inhomogeneous wavelet expansion starting from resolution `j0` up to resolution `j1`. Namely, it creates the wavelet expansion,

∑_{k = 0}^{2^{j_0} - 1} β_k φ_{j_0,k} + ∑_{j = j_0}^{j_1} ∑_{k = 0}^{2^j - 1} β_{j,k} ψ_{j,k}.

where (φ,ψ) denote the father and mother periodised Meyer wavelet functions and β_{j,k} denotes the mother wavelet coefficient at resolution j and location k and β_{k} denotes the father wavelet coefficients at resolution j=j0 and location k. The coefficients `beta` need to be ordered so that the first 2^\code{j0} elements correspond to father wavelet coefficients at resolution j=\code{j0} and the remaining elements correspond to the mother wavelet coefficients from resolution j=\code{j0} to j = log_2 n - 1. If the maximum resolution level j1 is not specified, the full wavelet expansion will be given.

## Value

A numeric vector of size n giving the wavelet function expansion.

`multiCoef`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```library(mwaved) # Make a noiseless doppler function n <- 2^8 x <- (1:n)/n y <- makeDoppler(n) # Determine the wavelet coefficients beta <- multiCoef(y) # plot three raw wavelet expansions truncating in each case at j1 = 3, 4 and 5 respectively plot(x, y, type = 'l', main = 'Doppler and wavelet projections at three different truncations') j0 <- 3 j1 <- 5 j <- j0:j1 lcols <- c(1, j - j0 + 2) ltys <- c(1, 1:length(j)) matlines(x, sapply(j, function(i) multiProj(beta, j1 = i)), type = 'l', col = lcols[-1]) legend("bottomright", legend = c("Signal", paste('j1 =', j)), col = lcols, lty =ltys) ```

### Example output

```
```

mwaved documentation built on July 13, 2017, 5:03 p.m.