Is it possible to define Fourier or wavelet transforms on DNA sequences?
First, a code for paper A measure of DNA sequence similarity by Fourier Transform with applications on hierarchical clustering. Journal of theoretical biology is available at GenomeDFT (github).
Then, on the transforms. Discrete Fourier transforms can be defined on other domains than vector spaces. There are Fourier transforms on groups, rings or finite fields. On the latter, it is commonly called a number-theoretic transform (NTT). This possibility entails to making sense of group/ring/field axioms for the nucleotides. For instance, it should be possible to use a finite field of characteristic four. Apparently weird mathematical structures could be interesting when considering (C,G) or (A,T)  patterns, or the redundancy in the codon/amino-acid correspondence.
Some papers mention the Ramanujan-Fourier transform (A Novel Method for Comparative Analysis of DNA Sequences by Ramanujan-Fourier Transform, apparently with the same first author of the paper you mentioned). With less structured transformations, it is also possible to evaluate periodicity, as for instance in Categorical spectral analysis of periodicity in nucleosomal DNA.
There also were some works on wavelets, either continuous or discrete. On the first flavor, by A. Arneodo and colleagues, and more recently in 2019 with
Wavelet-Based Genomic Signal Processing for Centromere Identification and Hypothesis Generation. With discrete wavelet transform, there is for instance Identification of exonic regions in DNA sequences using cross-correlation and noise suppression by discrete wavelet transform, Application of discrete wavelet transform for analysis of genomic sequences of Mycobacterium tuberculosis, Wavelet analysis of DNA sequences.
Finally, on the mappings. There is for instance Digital signal processing methods for biosequence comparison, 1991 or Mapping Equivalence for Symbolic Sequences: Theory and Applications, 2009.
As asked by Marcus, your actual purpose might direct you to an appropriate combination.