Mathematics is the foundational discipline for all sciences, engineering, and technology, and the pursuit of a normed division algebra of all finite dimensions represents a paramount mathematical objective. In the quest for a real three-dimensional, normed, associative division algebra, Hamilton discovered quaternions, constituting a non-commutative division algebra of quadruples. Subsequent investigations revealed the existence of only four division algebras over reals, each with dimensions 1, 2, 4, and 8. This study transcends such limitations by introducing generalized hypercomplex numbers extending across all dimensions, serving as extensions of traditional complex numbers. The space formed by these numbers constitutes a non-distributive normed division algebra extendable to all finite dimensions. The derivation of these extensions involves the definitions of two new $\pi$-periodic functions and a unified multiplication operation, designated as spherical multiplication, that is fully compatible with the existing multiplication structures. Importantly, these new hypercomplex numbers and their associated algebras are compatible with the existing real and complex number systems, ensuring continuity across dimensionalities. Most importantly, like the addition operation, the proposed multiplication in all dimensions forms an Abelian group while simultaneously preserving the norm. In summary, this study presents a comprehensive generalization of complex numbers and the Euler identity in higher dimensions, shedding light on the geometric properties of vectors within these extended spaces. Finally, we elucidate the practical applications of the proposed methodology as a viable alternative for expressing a quantum state through the multiplication of specified quantum states, thereby offering a potential complement to the established superposition paradigm. Additionally, we explore its utility in point cloud image processing.

In search of a real three-dimensional, normed, associative, division algebra, Hamilton discovered quaternions that form a non-commutative division algebra of quadruples. Later works showed that there are only four real division algebras with 1, 2, 4, or 8 dimensions. This work overcomes this limitation and introduces generalized hypercomplex numbers of all dimensions that are extensions of the traditional complex numbers. The space of these numbers forms non-distributive normed division algebra that is extendable to all finite dimensions. To obtain these extensions, we defined a unified multiplication, designated as scaling and rotative multiplication, fully compatible with the existing multiplication. Therefore, these numbers and the corresponding algebras reduce to distributive normed algebras for dimensions 1 and 2. Thus, this work presents a generalization of $\mathbb{C}$ in higher dimensions along with interesting insights into the geometry of the vectors in the corresponding spaces.

Fourier theory is a popular tool for analyzing various signals and interpreting their spectral contents. It finds applications in a wide variety of subject areas. However, some of the popular signals, such as sinusoidal signals, Dirac delta, signum, and unit step, fail to have a convergent Fourier representation (FR) in the conventional sense. Hence, it becomes imperative to utilize the distribution theory to understand and build a suitable representation for these signals. The signal processing and communication engineering literature does not explain these concepts clearly. As a result, many of the concepts of FR for signals that do not conform to the conventional derivations remain obscure to researchers. We attempt to bridge this gap and provide a comprehensive explanation regarding the existence of FR. Further, we have proposed a new linear space of Gauss–Schwartz (GS) functions and corresponding tempered superexponential (TSE) distributions. It is shown that the Fourier transform (FT) is an isomorphism on the GS space of test functions and hence by duality is an isomorphism on TSE distributions. The space GS is smallest in the sense that its dual space, a set of TSE distributions, is the largest linear space over which FT can be defined by duality.

This work presents a novel perfect reconstruction filter bank decomposition (PRFBD) for nonlinear and nonstationary time series and image data representation and analysis. The Fourier decomposition method (FDM), an adaptive approach wholly based on the Fourier representation, is shown to be a special case of the proposed PRFBD. The adaptive Fourier–Gauss decomposition (FGD) proposed in this work is a variation of the FDM, which is based on the FR and Gaussian filtering. Similarly, we also consider Butterworth filtering to develop adaptive Fourier–Butterworth decomposition (FBD). The proposed theory can decompose any signal (time series, image, or other data) into a set of the desired number of Fourier intrinsic band functions (FIBFs), which follow the amplitude-modulation and frequency-modulation (AM-FM) representations. A generic filterbank representation is also provided, where perfect reconstruction can be ensured for any given set of lowpass or highpass filters. We have performed an extensive analysis of both simulated and real-life data (COVID-19 pandemic, Earthquake and Gravitational waves) to demonstrate the efficacy of the proposed method. The resolution results in the time-frequency representation demonstrate that the proposed method is more promising than the state-of-the-art approaches.

Fourier theory is the backbone of the area of Signal Processing (SP) and Communication Engineering. However, Fourier series (FS) or Fourier transform (FT) do not exist for some signals that fail to fulfill a predefined set of Dirichlet conditions (DCs). We note a subtle gap in the explanation of these conditions as available in the popular signal processing literature. They lack a certain degree of explanation essential for the proper understanding of the same. For example, the original second Dirichlet condition is the requirement of bounded variations over one time period for the convergence of Fourier Series, where there can be at most infinite but countable number of maxima and minima, and at most infinite but countable number of discontinuities of finite magnitude. However, a large body of the literature replaces this statement with the requirements of finite number of maxima and minima over one time period, and finite number of discontinuities. The latter incorrectly disqualifies some signals from having valid FS representation. Similar problem holds in the description of DCs for the Fourier transform. Likewise, while it is easy to relate the first DC with the finite value of FS or FT coefficients, it is hard to relate the second and third DCs as specified in the signal processing literature with the Fourier representation as to how the failure to satisfy these conditions disqualifies those signals from having valid FS or FT representation.

This paper introduces Generalized Fourier transform (GFT) that is an extension or the generalization of the Fourier transform (FT). The Unilateral Laplace transform (LT) is observed to be the special case of GFT. GFT, as proposed in this work, contributes significantly to the scholarly literature. There are many salient contribution of this work. Firstly, GFT is applicable to a much larger class of signals, some of which cannot be analyzed with FT and LT. For example, we have shown the applicability of GFT on the polynomially decaying functions and super exponentials. Secondly, we demonstrate the efficacy of GFT in solving the initial value problems (IVPs). Thirdly, the generalization presented for FT is extended for other integral transforms with examples shown for wavelet transform and cosine transform. Likewise, generalized Gamma function is also presented. One interesting application of GFT is the computation of generalized moments, for the otherwise non-finite moments, of any random variable such as the Cauchy random variable. Fourthly, we introduce Fourier scale transform (FST) that utilizes GFT with the topological isomorphism of an exponential map. Lastly, we propose Generalized Discrete-Time Fourier transform (GDTFT). The DTFT and unilateral $z$-transform are shown to be the special cases of the proposed GDTFT. The properties of GFT and GDTFT have also been discussed.