THE theory of Fourier integrals arises out of the elegant pair of reciprocal formulae The Laplace Transform By David Vernon Widder. (Princeton Mathematical Series.) Pp. x + 406. (Princeton: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.

The Laplace Transform

We introduce an interpolation formula for functions in the Hardy space on the right-half plane and prove its convergence in norm and pointwise under very general condition. We also obtain an inverse formula for the Laplace transform from data on a finite interval.

Interpolation in the Hardy space

For a variety of non-spherical particles (oriented spheroids, cuboids, triangular prisms, and hexagonal prisms), analytical transform techniques are proposed to retrieve the particle size distribution (PSD) from measured absorption spectra. The absorption efficiency of particles is calculated using the anomalous diffraction theory (ADT). We find that for each type of non-spherical particles, there exists an ADT transform pair between the size distribution and the complex absorption spectrum, which provides the physical basis for solving the inverse problem. Furthermore, the relation between the size distribution and real absorption spectrum is established by using Gaver–Stehfest׳s method. The numerical calculations show that the use of extended precision instead of double precision arithmetic can produce more reliable results at the expense of computational efficiency. Also it is shown that a small Stehfest number (standing for truncation number) tends to enhance the anti-noise level of inversion.

Analytical inversion of the absorption spectrum to determine non-spherical particle size distribution

In this paper several weighted Lp-norm inequalities (p>1) for the Kontorovich–Lebedev–Fourier generalized convolutions are obtained With the help of these inequalities we consider a parabolic integro-differential equation of second order and study the boundedness of its solution in weighted Lp spaces Moreover, the boundedness of a scattered acoustic field in weighted Lp spaces is also obtained

Boundedness in weighted Lp spaces for the Kontorovich–Lebedev–Fourier generalized convolutions and applications

In this paper we introduce two generalized convolutions for the Fourier cosine, Fourier sine and Laplace integral transforms. Convolution properties and their applications to solving integral equations and systems of integral equations are considered.

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The Fourier–Laplace Generalized Convolutions and Applications to Integral Equations

In this paper, we obtain a new real-variable inverse formula for the Laplace transform and investigate its convergence rate.

A real-variable inverse formula for the Laplace transform

To study the boundedness of the acoustic field in $L_{p}(\mathbb {R}_{+} ; dx)$ and its asymptotic behavior, we introduce a new generalized convolution for the Kontorovich-Lebedev ($\mathcal {K}\mathcal {L}$) transform and the Fourier transforms, and a new representation of the acoustic field via this generalized convolution. Some properties of the new generalized convolution are obtained. Moreover, an analog of Watson’s theorem is established, in which we obtain the necessary and sufficient conditions for a class of generalized convolution transforms to be isomorphism-isometric.

Generalized Convolution for the Kontorovich-Lebedev, Fourier Transforms and Applications to Acoustic Fields

Integral equation of Toeplitz plus Hankel's type and parabolic equation related to the Kontorovich‐Lebedev‐Fourier generalized convolutions

We deal with several classes of integral transformations of the form 

$$f(x) \to D\int_{\mathbb{R}_ + ^2 } {\frac{1} {u}} \left( {e^{ - u\cosh (x + v)} + e^{ - u\cosh (x - v)} } \right)h(u)f(v)dudv,$$

, where D is an operator. In case D is the identity operator, we obtain several operator properties on Lp(ℝ+) with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on L2(ℝ+) and define the inversion formula. Further, for an other class of differential operators of finite order, we apply these transformations to solve a class of integro-differential problems of generalized convolution type.

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Trinh Tuan

Papers

A real-variable inverse formula for the Laplace transform

The Fourier–Laplace Generalized Convolutions and Applications to Integral Equations

Generalized Convolution for the Kontorovich-Lebedev, Fourier Transforms and Applications to Acoustic Fields

Integral equation of Toeplitz plus Hankel's type and parabolic equation related to the Kontorovich‐Lebedev‐Fourier generalized convolutions

On the Fourier cosine-Kontorovich-Lebedev generalized convolution transforms