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Recently, Lessig et al. [Appl Comput Harmon Anal. 2019;46:384-399] introduced the notion of bendlets; a shearlet-like system that is based on anisotropic scaling, translation, shearing and bending of a compactly supported generator. The theoretical framework of the bendlet transform is yet to be explored exclusively. Taking this opportunity, our aim is to investigate the mathematical properties of the bendlet transform such as the Rayleigh theorem, inversion formula, characterization of range and the pointwise convergence of the inversion formula. In continuation, we obtain the Heisenberg-type uncertainty principle and the Pitt's inequality for the bendlet transform. Subsequently, we employ the bendlet transform for obtaining the solutions of the wave and Laplace equations. Finally, we introduce the notion of quaternionic bendlet transform and also study its fundamental properties. [ABSTRACT FROM AUTHOR]

The present study is the first of its kind, aiming to explore the interface between the ridgelet and linear canonical transforms. To begin with, we formulate a family of linear canonical ridgelet waveforms by suitably chirping a one-dimensional wavelet along a specific direction. The construction of novel ridgelet waveforms is demonstrated via a suitable example supported by vivid graphics. Subsequently, we introduce the notion of linear canonical ridgelet transform, which not only embodies the classical ridgelet transform but also yields another new variant of the ridgelet transform based on the fractional Fourier transform. Besides studying all the fundamental properties, we also present an illustrative example on the implementation of the linear canonical ridgelet transform on a bivariate function. [ABSTRACT FROM AUTHOR]

In the article 'Convolution and product theorems for the special affine Fourier transform' [In: Nashed MZ, Li X, editors. Frontiers in orthogonal polynomials and q-series. World Scientific; 2018. p. 119–137], a convolution structure is presented in the realm of the special affine Fourier transform. In continuation of the study, we introduce a novel integral transform coined the special affine wavelet transform by combining the merits of the well-known special affine Fourier and wavelet transforms via the special affine convolution. The preliminary analysis encompasses the derivation of fundamental properties, Moyal's principle, inversion formula and range theorem. Subsequently, we obtain a mild extension of Heisenberg's uncertainty principle and also develop an analogue of Pitt's inequality for the special affine Fourier transform. In addition, we derive a Heisenberg-type uncertainty principle for the special affine wavelet transform. Finally, we extend the scope of the present study by introducing the notion of composition of special affine wavelet transforms. [ABSTRACT FROM AUTHOR]

For an efficient and directional representation of signals in higher dimensions, we propose the non-isotropic angular Stockwell transform in the context of time-frequency analysis. The proposed transform is aimed at rectifying the conventional Stockwell transform by employing an angular and scalable localized window which offers directional flexibility and thereby results in the multi-scale and directional analysis of signals in higher dimensions. The basic properties of the proposed transform such as orthogonality relation, reconstruction formula, derivation of the admissibility condition and characterization of the range are discussed using the machinery of operator theory and Fourier transforms. In addition, we introduce the discrete version of the non-isotropic angular Stockwell transform and establish a sufficient condition for the corresponding discrete family to be a frame in L 2 (R 2). Furthermore, some generalizations of the well-known Heisenberg-type inequalities are derived for the non-isotropic angular Stockwell transform in the Fourier domain. [ABSTRACT FROM AUTHOR]