Spectral synthesis on algebras of orthogonal polynomial series
Let K be a commutative hypergroup on \({\mathbb{N}}\cup \{0\}\) defined by a sequence of orthogonal polynomials \(P_ n\), and denote by \(L^ 1(K)\) the corresponding convolution algebra. The structure space \(\Delta (L^ 1(K))\) of \(L^ 1(K)\) can be identified with the set of all \(z\in {\mathbb{C}}\) such that \(\sup_{n}| P_ n(z)| \leq 1\). This class of hypergroups has been introduced and first studied by \textit{R. Lasser} [Rend. Math. Appl., VII. Ser. 3, 185-209 (1983; Zbl 0538.33010)]. Let \^K\(={\mathbb{R}}\cap \Delta (L^ 1(K))\) denote the set of all Hermitian elements in \(\Delta (L^ 1(K))\). Then \(L^ 1(K)\) is isomorphic to the pointwise multiplication algebra A(\^K) of functions f: \^K\(\to {\mathbb{C}}\), \(f=\sum^{\infty}_{n=1}\alpha_ n P_ n\), \(\| f\|_ 1=\sum^{\infty}_{n=1}| \alpha_ n| <\infty.\) The author proves that if K has polynomial growth, then the structure space of A(\^K) coincides with \(\hat K.\) His main result is a description of all primary ideals in \(L^ 1(K)\), i.e. the closed ideals in \(L^ 1(K)\) with a one-point hull in \(\hat K.\) In particular he obtains a criterion for points in \(\hat K\) to be spectral sets. These results considerably extend those of \textit{S. Wolfenstetter} [Arch. Math. 43, 364- 369 (1984; Zbl 0536.43013)]. Finally, the author gives applications to several examples, including the Chebyshev, Geronimus, Grinspun and q- ultraspherical polynomials.
scientific article; zbMATH DE number 3985877
scientific article