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In photoacoustic tomography, one is interested to recover the initial pressure distribution inside a tissue from the corresponding measurements of the induced acoustic wave on the boundary of a region enclosing the tissue. In the limited view problem, the wave boundary measurements are given on the part of the boundary, whereas in the full view problem, the measurements are known on the whole boundary. For the full view problem, there exist various fast and robust reconstruction methods. These methods give severe reconstruction artifacts when they are applied directly to the limited view data. One approach for reducing such artefacts is trying to extend the limited view data to the whole region boundary, and then use existing reconstruction methods for the full view data. In this paper, we propose an operator learning approach for constructing an operator that gives an approximate extension of the limited view data. We consider the behavior of a reconstruction formula on the extended limited view data that is given by our proposed approach. Approximation errors of our approach are analyzed. We also present numerical results with the proposed extension approach supporting our theoretical analysis.

For an absolutely continuous (integer-valued) r.v.X of the Pearson (Ord) family, we show that, under natural moment conditions, a Stein-type covariance identity of order holds (cf. [Goldstein and Reinert, J. Theoret. Probab. 18 (2005) 237-260]). This identity is closely related to the corresponding sequence of orthogonal polynomials, obtained by a Rodrigues-type formula, and provides convenient expressions for the Fourier coefficients of an arbitrary function. Application of the covariance identity yields some novel expressions for the corresponding lower variance bounds for a function of the r.v.X, expressions that seem to be known only in particular cases (for the Normal, see [Houdré and Kagan, J. Theoret. Probab. 8 (1995) 23-30]; see also [Houdré and Perez-Abreu, Ann. Probab. 23 (1995) 400-419] for corresponding results related to the Wiener and Poisson processes). Some applications are also given.

This work investigates the interplay between the Mellin transform and Lambert transforms to derive several novel results. In particular, we establish new inversion formulae for the Lambert transforms along with a Plancherel-type identity. Additionally, we explore the implications of these findings, highlighting their relevance to Salem’s equivalence and potential connections with the Riemann hypothesis.

Integral geometry problems involve finding a desired function from its integrals on a surface. These problems are closely intertwined with the generalized Radon transform, and obtaining an inversion formula for it is pivotal in solving integral geometry problems. The applications of integral geometry span various fields, including tomography, radar, and radiology. Particularly noteworthy is the recovery of a function from integrals over a parabola, which holds significance in reflection seismology. In our study, we concentrate on the transform that maps a real-valued smooth function with compact support to integrals over the paraboloid. This transform, along with its dual, can be expressed as convolutions of kernels and given functions, and we have derived inversion formulas based on their isometric properties.

We introduce the concept of a pseudoconvex set in an odd-dimensional Euclidean space. The inversion formula is obtained for the Radon transform in the case where the integrand is a piecewise continuous function defined on a pseudoconvex set. The result achieved is a generalization of a previously known property proved for smooth functions.

The problem of reconstruction of a function from spherical means is at the heart of several modern imaging modalities and other applications. In this paper we derive universal back-projection-type reconstruction formulas for recovering a function in arbitrary dimension from averages over spheres centered on the boundary of an arbitrarily shaped bounded convex domain with smooth boundary. Provided that the unknown function is supported inside that domain, the derived formulas recover the unknown function up to an explicitly computed integral operator. For elliptical domains the integral operator is shown to vanish and hence we establish exact inversion formulas for recovering a function from spherical means centered on the boundary of elliptical domains in arbitrary dimension. [PUBLICATION ABSTRACT]

In this paper, finite Meijer G‐transformation has been defined. A system of eigenfunctions is verified for the orthogonality condition. Operational calculus properties are shown. The classical concept of finite Meijer G‐transforms is studied in the present text. The construction of testing function spaces and their duals is established, and an application related to mathematical physics is illustrated.

This paper presents a new direct digital design method for discrete proportional integral derivative PID + filter (PIDF) controllers employed in DC-DC buck converters. The considered controller structure results in a proper transfer function which has the advantage of being directly implementable by a microcontroller algorithm. Secondly, it can be written as an Infinite Impulse Response (IIR) digital filter. Thirdly, the further degree of freedom introduced by the low pass filter of the transfer function can be used to satisfy additional specifications. A new design procedure is proposed, which consists of the conjunction of the pole-zero cancellation method with an analytical design control methodology based on inversion formulae. These two methods are employed to reduce the negative effects introduced by the complex poles in the transfer function of the buck converter while exactly satisfying steady-state specifications on the tracking error and frequency domain requirements on the phase margin and on the gain crossover frequency. The proposed approach allows the designer to assign a closed-loop bandwidth without constraints imposed by the resonance frequency of the buck converter. The response under step variation of the reference value, and the disturbance rejection capability of the proposed control technique under load variations are also evaluated in real-time implementation by using the Arduino DUE board, and compared with other methods.

This paper presents a comprehensive study of Plancherel’s theorem and inversion formulae for the Widder–Lambert transform, extending its scope to Lebesgue integrable functions, compactly supported distributions, and regular distributions with compact support. By employing the Plancherel theorem for the classical Mellin transform, we derive a corresponding Plancherel’s theorem specific to the Widder–Lambert transform. This novel approach highlights an intriguing connection between these integral transforms, offering new insights into their role in harmonic analysis. Additionally, we explore a class of functions that satisfy Salem’s equivalence to the Riemann hypothesis, providing a deeper understanding of the interplay between such equivalences and integral transforms. These findings open new avenues for further research on the Riemann hypothesis within the framework of integral transforms.