The exponential systems on for a discrete set possibly containing have been considered by Paley and Wiener in their book `R. E. A. C. Paley and N. Wiener. Fourier transforms in the complex domain. American Mathematical Society, New York, 1934.`

They called such systems the **nonharmonic Fourier series** to emphasize the distinction with the usual (harmonic) Fourier series when

In our work we develop a global version of the Fourier analysis adapted to spectral decompositions. If such spectral decomposition comes from a non-self-adjoint operator, it may lead to biorthogonal systems, ‘nonharmonic’ in the sense of Paley and Wiener. For the analysis of such quantizations we need, among other things, a notion of convolution, and we also analyse these in general Hilbert spaces.

More explanations to be added soon. For now just some of our papers on this subject:

- Our main first work on this subject:

Ruzhansky M., Tokmagambetov N.,, (2016) 2016 (12), 3548-3615. offprint (open access), arxiv, link**Nonharmonic analysis**of boundary value problems,**Int. Math. Res. Notices**

Further related works:

- Ruzhansky M., Tokmagambetov N.,
*Convolution, Fourier analysis, and distributions generated by Riesz bases*, arxiv - Delgado J., Ruzhansky M.,
*Fourier multipliers, symbols and nuclearity on compact manifolds,*, to appear, arxiv**J. Anal. Math.** - Ruzhansky M., Tokmagambetov N.,
*Wave equation for operators with discrete spectrum and irregular propagation speed*,, 226 (*Arch. Ration. Mech. Anal.**2017*), 1161-1207. offprint (open access), arxiv, link - Kanguzhin B., Ruzhansky M., Tokmagambetov N.,
*On convolutions in Hilbert spaces,*, 51**Funct. Anal. Appl.***(2017)*, 221-224. link (eng) link (rus) - Ruzhansky M., Tokmagambetov N.,
*Nonharmonic analysis of boundary value problems without WZ condition,***Math. Model. Nat. Phenom****.***,*12*(2017),*115-140. arxiv, link - Delgado J., Ruzhansky M., Tokmagambetov N.,
*Schatten classes, nuclearity and nonharmonic analysis on compact manifolds with boundary,*107**J. Math. Pures Appl.**,*(2017)*, 758-783. offprint (open access), arxiv, link - Dasgupta A., Ruzhansky M.,
*Eigenfunction expansions of ultradifferentiable functions and ultradistributions,*368**Trans. Amer. Math. Soc.**,*(2016),*8481-8498. arxiv, link - Dasgupta A., Ruzhansky M.,
*Eigenfunction expansions of ultradifferentiable functions and ultradistributions. II. Tensor representations*,, to appear, arxiv**Trans. Amer. Math. Soc.**