  In constructive quantum field theory (CQFT) it is customary to first
regularise the theory at finite UV and IR cut-off. Then one first removes the
UV cutoff using renormalisation techniques applied to families of CQFT's
labelled by finite UV resolutions and then takes the thermodynamic limit.
Alternatively, one may try to work directly without IR cut-off.
  More recently, wavelets have been proposed to define the renormalisation flow
of CQFT's which is natural as they come accompanied with a multi-resolution
analysis (MRA). However, wavelets so far have been mostly studied in the
non-compact case. Practically useful wavelets that display compact support and
some degree of smoothness can be constructed on the real line using Fourier
space techniques but explicit formulae as functions of position are rarely
available. Compactly supported wavelets can be periodised by summing over
period translates keeping orthogonality properties but still yield to rather
complicated expressions which generically lose their smoothness and position
locality properties.
  It transpires that a direct approach to wavelets in the compact case is
desirable. In this contribution we show that the Dirichlet-Shannon kernels
serve as a natural scaling function to define generalised orthonormal wavelet
bases on tori or copies of real lines respectively. These generalised wavelets
are smooth, are simple explicitly computable functions, display quasi-local
properties close to the Haar wavelet and have compact momentum supprt.
Accordingly they have a built-in cut-off both in position and momentum, making
them very useful for renormalisation applications.
