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We introduce a new family of invariant differential operators associated with$\lambda$-cosine and Funk-Radon transforms on Stiefel and Grassmann manifolds.These operators reduce the order of the $\lambda$-cosine transforms and yieldnew inversion formulas. Intermediate Funk-cosine transforms corresponding tointegration over matrices of lower rank are studied. The main tools are polardecomposition and Fourier analysis on matrix space.

The $\lambda$-Cosine Transforms, Differential Operators, and Funk Transforms on Stiefel and Grassmann Manifolds