Whether convolution followed by down sampling equals wavelet decomposition I know that wavelet decomposition equals convolution followed by down sampling by two.I tried to test this on mat lab on haar wavelet. I write the below code and compared with the default wavedec command in mat lab. The code I wrote is shown below. 

clear all
clc
h=[0.7071    0.7071];% Haar scaling filter
g=[-0.7071    0.7071];%Haar wavelet filter
x=[1:10];% Input 
h=fliplr(h);% h(-m)
g=fliplr(g);%g(-m)
s=conv(h,x);
s=downsample(s,2)
d=conv(g,x);
d=downsample(d,2)
[a L]=wavedec(x,1,'haar') % for comparison


After running this program I get s =  0.7071    3.5355    6.3639    9.1923   12.0207    7.0710.
d =  0.7071    0.7071    0.7071    0.7071    0.7071   -7.0710  and the wavedec out put as a = 2.1213    4.9497    7.7782   10.6066   13.4350   -0.7071   -0.7071   -0.7071   -0.7071   -0.7071. L =   5     5    10

I dont know why wavedec and my code gives different out puts, will any one help???
Simply put, no!


  I know that wavelet decomposition equals convolution followed by down sampling by two


No again. Some linear wavelet decompositions can be turned into a cascade of convolutions followed by downsampling. But wavelets are much more general. Their discrete implementations involve intricate properties in filters, downsampling, and levels.

If you just filter a signal  by some filter, and downsample, in general, you get almost nothing but aliasing.