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We introduce a family of generalized prolate spheroidal wave functions (PSWFs) of order $$-1,$$ - 1 , and develop new spectral schemes for second-order boundary value problems. Our technique differs from the differentiation approach based on PSWFs of order zero in Kong and Rokhlin (Appl Comput Harmon Anal 33(2):226–260, 2012); in particular, our orthogonal basis can naturally include homogeneous...
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