Sparse Fast DCT for Vectors with One-block Support

In this paper we present a new fast and deterministic algorithm for the inverse discrete cosine transform of type II that reconstructs the input vector $\mathbf{x}\in\mathbb{R}^{N}$, $N=2^{J-1}$, with short support of length $m$ from its discrete cosine transform $\mathbf{x}^{\widehat{\mathrm{II}}}=\mathbf{C}_N^{\mathrm{II}}\mathbf{x}$. The resulting algorithm has a runtime of $\mathcal{O}\left(m\log m\log \frac{2N}{m}\right)$ and requires $\mathcal{O}\left(m\log \frac{2N}{m}\right)$ samples of $\mathbf{x}^{\widehat{\mathrm{II}}}$. In order to derive this algorithm we also develop a new fast and deterministic inverse FFT algorithm that constructs the input vector $\mathbf{y}\in\mathbb{R}^{2N}$ with reflected block support of block length $m$ from $\widehat{\mathbf{y}}$ with the same runtime and sampling complexities as our DCT algorithm.