Determine the haar wavelet function is linear or nonlinear

Linearity can play at different levels:

• The Haar wavelet functions are piece-wise affine (constant), not linear.
• Any standard wavelet decomposition is linear, as it decomposes a signal into a sum of coefficients times a wavelet
• Wavelet approximations are often nonlinear, as they consist in choosing the $K$ best (largest) coefficients to represent the signal. And choosing the $K$ largest does not obey the linearity principle.

The wavelet function is preset and continuous signal defined in a time interval from 0 to N-1. The characteristics of this function is to be continuous (defined in any point). Important to remember that the function is applied with the convolution to your signal.

Then, the concept of linearity refers to, by definition, means that the output follows a linear trend with respect to the input (y=m*x+n).

Conclussion, wavelets functions are by definition continuous in time but not linear.

Hope this helps!

G.