definition of wavelet in $L_p$ space Function $\psi$ is called wavelet, if there is a dual  $\widetilde{\psi}$ such that a function $f \in L_2(R)$ can be decomposed as
$$
f(t)=\sum_{ j \in Z}\sum_{\nu \in Z} \langle f, \widetilde{\psi}_{j,\nu}\rangle \psi_{j,\nu}(t).
$$
How to define similarly wavelet for $f \in L_p$ space?
 A: Generally, $\psi$ is a wavelet if the family $\psi_{j,\nu}$ formed by the  dilations and translations of $\psi$ forms a basis of the space. Which leads one to ask: what is a basis? Most general notion is a Schauder basis; however it may be too general  in this context. The more common requirement is that $\psi_{j,\nu}$ form an unconditional basis of the space: that is, for every $f$ there is a unique sequence of coefficients $\psi_{j,\nu}$ such that the partial sums of $ \sum_{j,n} c_{j,\nu} \psi_{j,\nu}$ converge to $f$ in the norm, regardless of the order of summation.   
The computation of coefficients $c_{j,\nu}$ goes as follows: since $c_{0,0}$ is a bounded linear functional, it is represented by some element $\widetilde \psi$ of the dual space. To compute other $c_{j,\nu}$, we can appropriately translate/dilate the function $f$, reducing the computation to the case $j=0=\nu$. Via a change of variables, the translation/dilation operator can be pushed from $f$ to   $\widetilde \psi$. This results in the desired form of coefficients: they come from pairing $f$ with $\widetilde \psi_{j,\nu}$. 
Summary: same definition, with the understanding that $\psi\in L^p$, $\widetilde \psi\in L^q$, $1/p+1/q=1$, $1<p<\infty$.    
Reading suggestion: Wavelet bases of function spaces, notes by Vjeko Kovač.  
