In the mid 1980s it was conjectured that every bispectral meromorphic function $\psi (x,y)$ gives rise to an integral operator $K_{\psi }(x,y)$ which possesses a commuting differential operator. This has been verified by a direct computation for several families of functions $\psi (x,y)$ where the commuting differential operator is of order $\le 6$. We prove a general version of this conjecture for all self-adjoint bispectral functions of rank 1 and all self-adjoint bispectral Darboux transformations of the rank 2 Bessel and Airy functions. The method is based on a theorem giving an exact estimate of the second- and first-order terms of the growth of the Fourier algebra of each such bispectral function. From it we obtain a sharp upper bound on the order of the commuting differential operator for the integral kernel $K_{\psi }(x,y)$ leading to a fast algorithmic procedure for constructing the differential operator; unlike the previous examples its order is arbitrarily high. We prove that the above classes of bispectral functions are parametrized by infinite-dimensional Grassmannians which are the Lagrangian loci of the Wilson adelic Grassmannian and its analogs in rank 2.

中文翻译：

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积分算子，傅里叶代数的双谱和增长

在1980年代中期，人们推测每个双谱亚纯函数 $\psi （X，ÿ）$ 产生一个积分算子 ${ķ}_{\psi }（X，ÿ）$具有通勤的微分算子。通过对多个功能族的直接计算已验证了这一点。$\psi （X，ÿ）$ 通勤微分算子是有序的 $\le 6$。我们证明了该猜想的一般形式，适用于秩为1的所有自伴双谱函数以及秩为2的Bessel和Airy函数的所有自伴双谱Darboux变换。该方法基于一个定理，该定理给出了每个此类双谱函数的傅立叶代数增长的二阶和一阶项的精确估计。从中我们获得积分核通勤微分算子阶数的尖锐上限${ķ}_{\psi }（X，ÿ）$导致构造微分算子的快速算法过程；与前面的示例不同，其顺序任意高。我们证明了上述双谱函数的类别是由无限维的格拉斯曼主义者参数化的，后者是威尔逊·阿迪克斯格拉斯曼主义者及其类似物在第2级的拉格朗日基因座。