这是纯粹的哈尔缩放函数逼近$f(t)$

$$f(t)=\sum_{k=-\infty}^{\infty}a_k\phi(2^jt-k)$$ 在哪里 $$\phi(t) = \begin{cases}1 \quad; 0 \leq t < 1,\\0\quad;\mbox{otherwise.}\end{cases}$$ 是哈尔缩放函数，$j$缩放参数，$k$平移参数和系数 $$a_k=\int f(t)\phi(2^jt-k)dt$$ 所以在我们的情况下，如果$f(t)=t$， 为了$k\in \mathbb{Z}$并且对于$j=0$我们必须计算$a_k=\int f(t)\phi(t-k)dt$. 我们知道$\phi (t-k)=1$为了$t\in [k,k+1)$其他地方为零。所以我们只整合$[k,k+1)$. $$a_k=\int_{k}^{k+1} t \phi(t-k)dt=\frac{2k+1}{2}$$ 因此我们有 $$f(t)=\frac12\sum_{k=-\infty}^{\infty}(2k+1)\phi(t-k)$$ 以下脚本绘制了 haar 近似与给定函数的关系$f(x)=x$

import numpy as np
import matplotlib.pyplot as plt

def init(x,j,k):
    y=np.piecewise(x,[x < 0, x > 1],[lambda x: 0, lambda x: 0, lambda x:1]) 
    t=(x+k)/2**j
    ii=np.where(y==1)
    return y[ii], t[ii]

a=-4 #lower limit of sum
b=4 #upper limit of sum
j=0
k=np.arange(a*2**j,b*2**j)
y=(2*k+1)/2 # the coefficients a_k
T=np.arange(0,2,0.01)
[p,dummy]=init(T*2**j,j,0)
phi=np.ones((len(k),len(p)))
t=np.zeros((len(k),len(p)))

for i in range (1,len(k)+1):
    [p,t1]=init(T*2**j,j,i-1)
    t[i-1,:]=t1+a

for l in range (1,len(k)+1):
    plt.plot(t[l-1,:],2**j*y[l-1]*phi[l-1,:],'r') 

哈尔近似$f(x)=cos(x)$为了$j=2$

感谢 Jan。我完成了我的实施工作。
下面的代码比较：Haar vs Fourier vs Chebyshev。

from __future__ import division
from mpmath import *

# --------- Haar wavelet approximation of a function
# algorithm from : http://fourier.eng.hmc.edu/e161/lectures/wavelets/node5.html
# implementation only handle [0,1] for the moment: scaling and wavelet fcts need to be periodice

phi = lambda x : (0 <= x < 1) #scaling fct
psi = lambda x : (0 <= x < .5) - (.5 <= x < 1) #wavelet fct
phi_j_k = lambda x, j, k : 2**(j/2) * phi(2**j * x - k)
psi_j_k = lambda x, j, k : 2**(j/2) * psi(2**j * x - k)

def haar(f, interval, level):
    c0 = quadgl(  lambda t : f(t) * phi_j_k(t, 0, 0), interval  )

    coef = []
    for j in xrange(0, level):
        for k in xrange(0, 2**j):
                djk = quadgl(  lambda t: f(t) * psi_j_k(t, j, k), interval  )
                coef.append( (j, k, djk) )

    return c0, coef

def haarval(haar_coef, x):
    c0, coef = haar_coef
    s = c0 * phi_j_k(x, 0, 0)
    for j, k ,djk in coef:
            s += djk * psi_j_k(x, j, k)
    return s

# --------- to plot an Haar wave
interval = [0, 1]
plot([lambda x : phi_j_k(x,1,1)],interval)

# ---------- main
# below is code to compate : Haar vs Fourier vs Chebyshev

nb_coeff = 5
interval = [0, 1] # haar only handle [0,1] for the moment: scaling and wavelet fcts need to be periodice

fct = lambda x : x

haar_coef = haar(fct, interval, nb_coeff)
haar_series_apx = lambda x : haarval(haar_coef, x)

fourier_coef = fourier(fct, interval, nb_coeff)
fourier_series_apx = lambda x: fourierval(fourier_coef, interval, x)
chebyshev_coef = chebyfit(fct, interval, nb_coeff)
chebyshev_series_apx = lambda x : polyval(chebyshev_coef, x)


print 'fourier %d chebyshev %d haar %d' % ( len(fourier_coef[0]) + len(fourier_coef[1]),len(chebyshev_coef), 1 + len(haar_coef[1]))
print 'error:'
print 'fourier', quadgl(  lambda x : abs( fct(x) - fourier_series_apx(x) ), interval  )
print 'chebyshev', quadgl(  lambda x : abs( fct(x) - chebyshev_series_apx(x) ), interval  )
print 'haar', quadgl(  lambda x : abs( fct(x) - haar_series_apx(x) ), interval  )

plot([fct, fourier_series_apx, chebyshev_series_apx, haar_series_apx], interval)