It is already known that a curve estimated via the discrete cosine transform (DCT) always passes through all the measured points without using imaginary numbers in the DCT coefficients, unlike the discrete Fourier transform. Moreover, owing to its character, the DCT can be used instead of the nonlinear least-squares method to express various theoretical curves. Because the DCT is a kind of Fourier transform, there is a possibility that the DCT could be employed to draw electron density distribution maps of crystals. If so, the probability that the DCT could be used to investigate the internal structure of materials by analysing the theoretical curves would increase. This article reports an attempt to draw the electron density distribution maps of the Mg₃BN₃ low-pressure phase [Mg₃BN₃(L)] by using the DCT in order to confirm the utility of the DCT for analysing the internal structure of materials. It is found that the DCT can provide mirror-symmetric electron density distribution projection maps and a modified DCT can be used to calculate whole standard electron density distribution projection maps for the surface plane of the unit cell. Moreover, a real crystal structure that has a centre of symmetry can be determined by the DCT by transforming a 1/4 part of the mirror-symmetric electron density distribution projection map.

中文翻译：

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离散余弦变换计算电子密度分布投影图的新研究

众所周知，与离散傅立叶变换不同，通过离散余弦变换 (DCT) 估计的曲线总是通过所有测量点，而无需在 DCT 系数中使用虚数。此外，由于其特性，DCT 可以代替非线性最小二乘法来表达各种理论曲线。由于 DCT 是一种傅里叶变换，因此有可能将 DCT 用于绘制晶体的电子密度分布图。如果是这样，DCT 可用于通过分析理论曲线来研究材料内部结构的可能性将增加。本文报道了绘制 Mg ₃ BN ₃低压相 [Mg₃ BN ₃ (L)] 通过使用 DCT 来确认 DCT 在分析材料内部结构方面的效用。发现DCT可以提供镜像对称的电子密度分布投影图，并且可以使用修改的DCT来计算晶胞表面平面的整体标准电子密度分布投影图。此外，通过变换镜像对称电子密度分布投影图的 1/4 部分，可以通过 DCT 确定具有对称中心的真实晶体结构。