Due to the extra degrees of freedom, special affine Fourier transform (SAFT) has achieved a respectable status within a short span and got versatile applicability in the areas of signal processing, image processing, sampling theory, quantum mechanics. However, due to its global kernel, SAFT fails to obtain local information of non-transient signals. To overcome this, in this paper, we introduce the concept of novel special affine wavelet transform (NSAWT) and extend key harmonic analysis results to NSAWT analogous to those for the wavelet transform. We first establish some fundamental properties including Moyal’s principle, Inversion formula and the range theorem. Some Heisenberg type inequalities and Pitt’s inequality are established for SAFT and consequently Heisenberg uncertainty principle is derived for NSAWT.

中文翻译：

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新颖的特殊仿射小波变换和相关的不确定性原理

由于额外的自由度，特殊仿射傅里叶变换（SAFT）在短时间内取得了可观的地位，并在信号处理、图像处理、采样理论、量子力学等领域得到了广泛的应用。然而，由于其全局内核，SAFT 无法获取非瞬态信号的局部信息。为了克服这个问题，在本文中，我们引入了新的特殊仿射小波变换 (NSAWT) 的概念，并将关键谐波分析结果扩展到类似于小波变换的 NSAWT。我们首先建立了一些基本性质，包括Moyal原理、反演公式和范围定理。为SAFT建立了一些Heisenberg型不等式和Pitt不等式，由此推导出NSAWT的Heisenberg不确定性原理。