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Register a new userThe V-line transform with some generalizations and cone differentiation
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Mohammad Javad Latifi Jebelli
Publication date
2016
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English
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The paper studies various properties of the V-line transform (VLT) in the plane and conical Radon transform (CRT) in $mathbb{R}^n$. VLT maps a function to a family of its integrals along trajectories made of two rays emanating from a common point. The CRT considered in this paper maps a function to a set of its integrals over surfaces of polyhedral cones. These types of operators appear in mathematical models of single scattering optical tomography, Compton camera imaging and other applications. We derive new explicit inversion formulae for VLT and CRT, as well as proving some previously known results using more intuitive geometric ideas. Using our inversion formula for VLT, we describe the range of that transformation when applied to a fairly broad class of functions and prove some support theorems. The efficiency of our method is demonstrated on several numerical examples. As an auxiliary result that plays a big role in this article, we derive a generalization of the Fundamental Theorem of Calculus, which we call Cone Differentiation Theorem.
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We study the inverse problem of recovering a vector field in $mathbb{R}^2$ from a set of new generalized $V$-line transforms in three different ways. First, we introduce the longitudinal and transverse $V$-line transforms for vector fields in $mathbb{R}^2$. We then give an explicit characterization of their respective kernels and show that they are complements of each other. We prove invertibility of each transform modulo their kernels and combine them to reconstruct explicitly the full vector field. In the second method, we combine the longitudinal and transverse V-line transforms with their corresponding first moment transforms and recover the full vector field from either pair. We show that the available data in each of these setups can be used to derive the signed V-line transform of both scalar component of the vector field, and use the known inversion of the latter. The final major result of this paper is the derivation of an exact closed form formula for reconstruction of the full vector field in $mathbb{R}^2$ from its star transform with weights. We solve this problem by relating the star transform of the vector field to the ordinary Radon transform of the scalar components of the field.
We show how polynomial mappings of degree k from a union of disjoint intervals onto [-1,1] generate a countable number of special cases of a certain generalization of the Chebyshev Polynomials. We also derive a new expression for these generalized Chebyshev Polynomials for any number of disjoint intervals from which the coefficients of x^n can be found explicitly in terms of the end points and the recurrence coefficients. We find that this representation is useful for specializing to the polynomial mapping cases for small k where we will have algebraic expressions for the recurrence coefficients in terms of the end points. We study in detail certain special cases of the polynomials for small k and prove a theorem concerning the location of the zeroes of the polynomials. We also derive an expression for the discriminant for the case of two intervals that is valid for any configuration of the end points.
First some definite integrals of W. H. L. Russell, almost all with trigonometric function integrands, are derived, and many generalized. Then a list is given in Russell-style of generalizations of integral identities of Amdeberhan and Moll. We conclude with a brief and noncomprehensive description of directions for further investigation, including the significant generalization to elliptic functions.
Let $W$ denote a matrix $A_2$ weight. In this paper, we implement a scalar argument using the square function to deduce square-function type results for vector-valued functions in $L^2(mathbb{R},mathbb{C}^d)$. These results are then used to study the boundedness of the Hilbert transform and Haar multipliers on $L^2(mathbb{R},mathbb{C}^d)$. Our proof shortens the original argument by Treil and Volberg and improves the dependence on the $A_2$ characteristic. In particular, we prove that the Hilbert transform and Haar multipliers map $L^2(mathbb{R},W,mathbb{C}^d)$ to itself with dependence on on the $A_2$ characteristic at most $[W]_{A_2}^{frac{3}{2}} log [W]_{A_2}$.
We construct a family of second-order linear difference equations parametrized by the hypergeometric solution of the elliptic Painleve equation (or higher-order analogues), and admitting a large family of monodromy-preserving deformations. The solutions are certain semiclassical biorthogonal functions (and their Cauchy transforms), biorthogonal with respect to higher-order analogues of Spiridonovs elliptic beta integral.
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