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The discrete cosine transform. (English) Zbl 0939.42021
In this paper the author derives the Discrete Cosine Transform (DCT) bases as eigenvectors of a symmetric second-difference matrix with certain boundary conditions. The type of boundary condition (Dirichlet or Neumann, centered at a meshpoint or a midpoint) determines the applications that are appropriate for each transform. The noteworthy observation is that all these “eigenvectors of cosines” come from simple and familiar matrices. More specific, each matrix contains a circular block between the second and before the last row, constructed with \((-1,2,-1)\) around the diagonal. The first and last row are different for each particular boundary condition.

MSC:
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
94A11 Application of orthogonal and other special functions
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
65T60 Numerical methods for wavelets
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