Hilbert spaces play a prominent role in various fields of mathematics. An orthonormal basis of such a space is called a Hilbert basis. The purpose of this blog is to illustrate a very clasic and basic Hilbert basis – the Haar basis.

Let be a Hilbert basis of equipped with the standard scalar product . Hence, every can be written in a unique way as

where and .

A classic Hilbert basis consists of Haar functions that are supported on . They are defined using the Haar wavelet:

The Haar basis consists then of rescaled (by ) versions of shifted by ,

together with the constant function . (Note: the constant function has to be added since we consider the interval and not ). In R this looks like:

```
haar_mother <- function(x){
(x >0 & x <= 0.5) - (x >0.5 & x <= 1)
}
haar <- function(x,j,k){
2^(j/2) * haar_mother(2^j*x-k)
}
```

#### Animation of the Haar basis

```
j_max <- 3 # maximal depth
n_max <- 10 # resolution of grid
df <- data.frame(x=numeric(),
y=numeric(),
id=numeric())
x <- (1: 2^n_max)/2^n_max
id<-1
for (j in 0:j_max){
for (k in 0:(2^j-1))
{
df_new<- data.frame(x=x,
y=haar(x,j,k),
id=id)
df <- bind_rows(df, df_new)
id <- id+1
}
}
ggplot(df, aes(x=x, y=y)) +
geom_step()+
transition_states(
id,
transition_length = 2,
state_length = 5
) +
labs(title = 'Illustration of Haar basis') +
ease_aes('sine-in-out')
```

#### Approximation via Haar basis

Now, every function can be written as

The coefficients are called the Haar Wavelet coefficients. Let us calculate them in the discrete setting. We give us a mesh and values and some function :

```
j_max <- 12
n_max <- 13 # maximal resolution
delta <- 2^{-n_max}
x <- (1: 2^n_max)/2^n_max
y <- x* sin(1/x)
alpha <- list() # list of Haar coefficients
for (j in 0:j_max){
alpha[[j+1]] <- rep(0, 2^j)
for (k in 0:(2^j-1))
{
alpha[[j+1]][k+1] <-sum(haar(x,j,k)*y)*delta # approximation of scalar product
}
}
y_approx <- rep(0, length(y)) # approximated values of y
#### Calculating the approximations for different values of j
df <- data.frame(x=numeric(),
y=numeric(),
id=numeric())
y_approx <- mean(y) # this is the contribution of the constant function
for (j in 0:j_max){
for (k in 0:(2^j-1))
{
y_approx <- y_approx + alpha[[j+1]][k+1]*haar(x,j,k)
}
df_new <- data.frame(x=x,
y=y_approx,
id=j)
df <- bind_rows(df, df_new)
}
```

## Animation of convergence

```
ggplot(df, aes(x=x, y=y)) +
geom_step()+
transition_states(
id,
transition_length = 2,
state_length = 1
) +
labs(title = "Haar basis approximation of x sin(1/x). Depth: {closest_state}") +
ease_aes('sine-in-out')
```

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