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In this paper, we recall the main results of the Fourier transform on the quaternionic Heisenberg group, then we introduce the notion of the Gabor transform on this group and study some of its properties. Finally, we prove some uncertainty principles associated with this transform. [ABSTRACT FROM AUTHOR]

In order to calculate the Wiener integrals for functionals on Wiener space, one can usually apply the change of variable theorem. But, there are many functionals that are difficult or impossible to calculate even when using the change of variable formula. In order to solve this problem, we establish an evaluation formula via the Fourier-type functionals on Wiener space. We then present various examples to which our evaluation formula can be applied and with the corresponding numerical approximations. [ABSTRACT FROM AUTHOR]

In this paper, a series of new high-order numerical approximations toα-th Caputo derivatives (0<α<2) is derived based on a compound of shift operators and high-order approximations to Riemann–Liouville derivatives. The convergence order is independent of the derivative orderα, rather than the previous error estimates. Several numerical examples including the Caputo-type advection–diffusion equation are displayed, which support the derived numerical schemes. [ABSTRACT FROM PUBLISHER]

Periodic wavelet frames have gained considerable popularity in recent years, primarily due to their substantiated applications in diverse and widespread fields of science and engineering. In this article, we introduce the concept of periodic wavelet frame on local fields of positive characteristic and show that under weaker conditions the periodization of any wavelet frame constructed by the unitary extension principle with dilation factor 𝔭−1is a periodic wavelet frame on local fields. Moreover, based on the mixed extension principle and Fourier-based techniques of the wavelet frames, we present an explicit method for a pair of dual periodic wavelet frames on local fields of positive characteristic. [ABSTRACT FROM PUBLISHER]

Quaternion-valued functions have been used as a model for colour images and have recently been studied using various Fourier-type transforms. We develop some fundamental wavelet theory for quaternionic signals using the Fourier kernel introduced by Brackx, De Schepper, and Sommen in [1]. We present several analogs to classical wavelet theory, such as the quadrature mirror filter condition. We also include necessary design conditions for a wavelet basis to have desired regularity and sufficient design conditions which will guarantee compact support. Due to the difficulty in constructing orthogonal wavelet bases, we present some theory for biorthogonal wavelet bases and construct a few examples. [ABSTRACT FROM AUTHOR]

We consider the problem of reconstructing, from the interior data u(x, 1), a function u satisfying a nonlinear elliptic equation [image omitted] The problem is ill-posed. Using the method of Green functions, the method of Fourier transforms, and the method of truncated high frequencies, we shall regularize the problem. Error estimate is given. [ABSTRACT FROM AUTHOR]