This paper presents some recent results concerning spectral analysis and spectral synthesis on polynomial hypergroups. The main results show that both discrete spectral analysis and discrete spectral synthesis hold for any polynomial hypergroup.
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Székelyhidi, L. Spectral Analysis and Spectral Synthesis on Polynomial Hypergroups. Monatsh. Math. 141, 33–43 (2004). https://doi.org/10.1007/s00605-002-0003-4