Inversion Formula for Continuous Multifractals

In an earlier paper [MR] the authors introduced the inverse measure μ[dagger](dt) of a given measure μ(dt) on [0, 1] and presented the ‘inversion formula’ f[dagger](α)=αf(1/α) which was argued to link the respective multifractal spectra of μ and μ[dagger]. A second paper [RM2] established the formula under the assumption that μ and μ[dagger] are continuous measures.Here, we investigate the general case which reveals telling details of interest to the full understanding of multifractals. Subjecting self-similar measures to the operation μ[map ]μ[dagger] creates a new class of discontinuous multifractals. Calculating explicitly we find that the inversion formula holds only for the ‘fine multifractal spectra’ and not for the ‘coarse’ ones. As a consequence, the multifractal formalism fails for this class of measures. A natural explanation is found when drawing parallels to equilibrium measures. In the context of our work it becomes natural to consider the degenerate Holder exponents 0 and ∞.

Let B B be a real separable Banach space, and let μ \mu be a probability measure on B ( B ) \mathcal {B}(B) , the Borel sets of B B . The characteristic functional (Fourier transform) ϕ \phi of μ \mu , defined by ϕ ( y ) = ∫ B exp ⁡ { i ( y , x ) } d μ ( x ) \phi (y) = \int _B {\exp \{ i(y,x)\} d\mu (x)\;} for y ∈ B ∗ y \in {B^\ast } (the topological dual of B B ), uniquely determines μ \mu . In order to determine μ \mu on B ( B ) \mathcal {B}(B) , it suffices to obtain the value of ∫ B G ( s ) d μ ( s ) \int _B {G(s)d\mu (s)} for every real-valued bounded continuous function G G on B B . Hence an inversion formula for μ \mu in terms of ϕ \phi is obtained if one can uniquely determine the value of ∫ B G ( s ) d μ ( s ) \int _B {G(s)d\mu (s)} for all real-valued bounded continuous functions G G on B B in terms of ϕ \phi and G G . The main efforts of this paper will be to prove such inversion formulae of various types. For the Orlicz space E α {E_\alpha } of real sequences we establish inversion formulae (Main Theorem II) which properly generalize the work of L. Gross and derive as a corollary the extension of the Main Theorem of L. Gross to E α {E_\alpha } spaces (Corollary 2.2.12). In Part I we prove a theorem (Main Theorem I) which is Banach space generalization of the Main Theorem of L. Gross by reinterpreting his necessary and sufficient conditions in terms of convergence of Gaussian measures. Finally, in Part III we assume our Banach space to have a shrinking Schauder basis to prove inversion formulae (Main Theorem III) which express the measure directly in terms of ϕ \phi and G G without the use of extension of ϕ \phi as required in the Main Theorems I and II. Furthermore this can be achieved without using the Lévy Continuity Theorem and we hope that one can use this theorem to obtain a direct proof of the Lévy Continuity Theorem.

The Pick's theorem is one of the rare gems of elementary mathematics because this is a very innocent sounding hypothesis imply a very surprising conclusion (Bogomolny 1997). Yet the statement of the theorem can be understood by a fifth grader. Call a polygon a lattice polygon if the co-ordinates of its vertices are integers. Pick's theorem asserts that the area of a lattice polygon P is given by A(P) = I(P) + B(P) / 2 - 1 = V(P) - B(P) / 2 - 1 where I(P), B(P) and V(P) are the number of interior lattice points, the number of boundary lattice points and the total number of lattice points of P respectively. It is worth to mention that the I(P) (understand like digital area) is digital mapping standard in USA since decade (Morrison, J. L. 1988 and 1989). Because the Pick's theorem was first published in 1899 therefore our planned presentation had timing its 100 anniversary. Currently it has greater importance than realized heretofore because of the Pick's theorem forms a connection between the old Euclidean and the new digital (discrete) geometry. During this long period lots of proof had been made of Pick's theorem and many trial of its generalization from simple polygons towards complex polygon networks, moreover tried to extend it to the direction of 3D geometrical objects as well. It is also turned out that nowadays the inverse Pick's formulas comes to the front instead of the original ones, consequently of powerful spreading the digital geometry and mapping. Today the question is not the old one: how can we produce traditional area without co-ordinates, using only inside points and boundary points. Just on the contrary: how is it possible to simply determine digital boundary and digital area (namely the number of boundary points and inside points) using known co-ordinates of vertices. The inverse formulas are: B(P)=ΣGCD (AX, AY, AZ) (1D Pick's theorem) and I(P)=A(P)-B(P)/2+1 (2D Pick's theorem) where GCD is the Great Common Divisor of the co-ordinate differences of two-two neighboring vertices. The our main object is not these formulas to present, but we desire to show that the Pick's theorem (after adequate redrafting) indeed valid for every spatial triangle which are determined by three arbitrary points of a 3D lattice. The original planar theorem is only a special case of it. However if it is true then its valid not only for triangles but all irregular polygons also which are lying in space and have its vertices in spatial lattice points. Finally if the extended Pick's theorem is true for all face of a lattice polyhedron then it is true for total surface as well. Consequently we developed so simple and effective algorithms which solve enumeration tasks without the time- and memory-wasting immediate computing. These algorithms make possible that using the vertex-co-ordinate list and the topological description of a convex or non-convex polyhedron (cube, prism, tetrahedron etc.) getting answer many elementary questions. For example, how many vaxels can be found on the complex surface of a polyhedron, how many on its edges or on its individual faces. We succeeded to extend our results also to the surface of non-cornered geometric objects (circle, sphere, cylinder, cone, ellipsoid etc.), but anyway, this have to be object of another presentation.

Inverse Measures, the Inversion Formula, and Discontinuous Multifractals

In this article, an inversion formula is obtained for the spherical transform which integrates functions, defined on the unit sphere \(S^{2}\), on circles. The inversion formula is for the case where the circles of integration are obtained by intersections of \(S^{2}\) with hyperplanes passing through a common point \(\overline{a}\) strictly inside \(S^{2}\). In particular, this yields inversion formulas for two well-known special cases. The first inversion formula is for the special case where the family of circles of integration consists of great circles; this formula is obtained by taking \(\overline{a} = 0\). The second inversion formula is for the special case where the circles of integration pass through a common point \(p\) on \(S^{2}\); this formula is obtained by taking the limit \(\overline{a}\rightarrow p\).

B.D. Smith (ibid., vol.MI-4, p.15-25, 1985; Opt. Eng., vol.29, p.524-34, 1990) and P. Grangeat (These de doctorat, 1987; Lecture Notes in Mathematics 1497, p.66-97, 1991) derived a cone-beam inversion formula that can be applied when a nonplanar orbit satisfying the completeness condition is used. Although Grangeat's inversion formula is mathematically different from Smith's one, they have similar overall structures to each other. The contribution of the present paper is two-fold. First, based on the derivation of Smith, the authors point out that Grangeat's inversion formula and Smith's one can be conveniently described using a single formula (the Smith-Grangeat inversion formula) that is in the form of space-variant filtering followed by cone-beam back projection. Furthermore, the resulting formula is reformulated for data acquisition systems with a planar detector to obtain a new reconstruction algorithm. Second, the authors make two significant modifications to the new algorithm to reduce artifacts and numerical errors encountered in direct implementation of the new algorithm. As for exactness of the new algorithm, the following fact can be stated. The algorithm based on Grangeat's intermediate function is exact for any complete orbit, whereas that based on Smith's intermediate function should be considered as an approximate inverse excepting the special case where almost every plane in 3D space meets the orbit. The validity of the new algorithm is demonstrated by simulation studies.

Given n disjoint intervals on together with n functions , , and an matrix , the problem is to find an L2 solution , , to the linear system , where , is a matrix of finite Hilbert transforms with defined on , and is a matrix of the corresponding characteristic functions on . Since we can interpret , as a generalized multi‐interval finite Hilbert transform, we call the formula for the solution as “the inversion formula” and the necessary and sufficient conditions for the existence of a solution as the “range conditions”. In this paper we derive the explicit inversion formula and the range conditions in two specific cases: a) the matrix Θ is symmetric and positive definite, and; b) all the entries of Θ are equal to one. We also prove the uniqueness of solution, that is, that our transform is injective. In the case a), that is, when the matrix Θ is positive definite, the inversion formula is given in terms of the solution of the associated matrix Riemann–Hilbert Problem. In the case b) we reduce the multi interval problem to a problem on n copies of and then express our answers in terms of the Fourier transform. We also discuss other cases of the matrix Θ.

The use of the repeated Cauchy principal value affords greater facility in the application of inversion formulae involving characteristic functions. Formula (2) below is especially useful in obtaining the inversion formula (1) for the distribution of the ratio of linear combinations of random variables which may be correlated. Formulae (1), (10), (12) generalize the special cases considered by Cramer [2], Curtiss [4], Geary [6], and are free of some restrictions they impose. The results are further generalized in section 6, where inversion formulae are given for the joint distribution of several ratios. In section 7, the joint distribution of several ratios of quadratic forms in random variables $X_1, X_2,\cdots,X_n$ having a multivariate normal distribution is considered.

In two dimensions, the exponential X-ray transform has been well-studied due to its applications of correcting attenuation effects in single photon emission computed tomography (SPECT). Explicit inversion formulas have been known for over 15 years. The three-dimensional (3D) case has not been as thoroughly examined, and inversion formulas are available for only a few of the wide range of possible 3D geometries. The rotating slant-hole (RSH) SPECT geometry is a special case for which no inversion formula has yet appeared. This paper presents a general inversion formula for the 3D exponential X-ray transform using a Neumann series. The method applies to any geometry but convergence of the series depends on the exponential scalar and the size of the region-of-interest. The derivation is presented in the context of the RSH SPECT geometry. Results from computer simulations are given.

The windowed ray transform is the generalization of the “Analytic-Signal Transform” which was developed to extend arbitrary functions from ℝ n ${\mathbb{R}^{n}}$ to ℂ n ${\mathbb{C}^{n}}$ in a natural way. The X-ray transform is also a special case of this transform. Similar to the X-ray transform, the problem of inverting this transform is overdetermined. Hence it is possible to prove the existence of several inversion formulas. Kaiser obtained one such inversion formula in 1993. We present several inversion formulas here. One of them is similar to that of Kaiser, but it requires a weaker condition. The others are new.

Here sn x is an abbreviation for sin (=rx/2). This paper gives other conditions for the validity of these identities. The previous conditions permitted qk to have various types of discontinuity. The present paper is concerned with smooth functions; however, the growth at 0 and oo is permitted to be greater than before. Theorems 1, 2, and 3 of the previous paper together with Theorems 2 and 3 of this paper form a fairly complete elementary theory of these identities. The proofs given here do not depend on the previous paper. The results of this paper hinge on the possibility of defining the Fourier sine transform for functions which do not vanish at infinity. Theorem 1 below shows that this is possible merely by employing summability. It is to be noted that Theorem 1 is not true, as it stands, for the cosine transform. For example, the cosine transform of any constant evaluated by such a definition would vanish. Hence the inversion formula could not apply. A theory of generalized Fourier

A generalization of Cooke's integral inversion formula with application to remote-sensing theory

This paper presents a contractive operator view on the inversion formula for finite Toeplitz operator matrices due to Gohberg-Heinig. The general setting that will be used involves a Hilbert space operator T and a contraction A such that the compression of T - A*TA to the orthogonal complement of the defect space of A is the zero operator. For such an operator T the analogue of the Gohberg-Heinig inversion formula is obtained. The main results are illustrated on various special cases, including Toeplitz plus Hankel operators and model operators.

1. In the literature on the Laplace transformation much attention has been devoted to inversion formulae, and a considerable number of these is known. In the theory, they often form the basis of representation theorems, that is, statements of necessary and sufficient conditions under which a function is a Laplace transform, or a Laplace transform of a function of a certain class; and in the applications inversion formulae are often used at the last stage of the solution of a problem by means of the Laplace transformation. While many individual inversion formulae have been investigated, it seems that no general principle has developed that will yield all of them. Yet, there is such a principle, and it can be formulated briefly (if somewhat inaccurately) as follows. If there is a singular integral whose kernel N(u, t; k) can be interpreted as the result of a linear operation on e-8u so that N u Lk, t[e5u], then Lk,t is an inversion operator for the Laplace transformation. The extension of this principle to the Laplace-Stieltjes transformation is imnmediate, and it is also clear that, with small changes, the principle applies to other linear functional transformations, notably integral transformations. The principle is not new. In special cases it has been used repeatedly to prove individual inversion formulae. However, it does not appear to have been formulated in a general manner, nor does it seem to have been exploited for discovering inversion operators except possibly for Stieltjes' discovery of the inversion by means of derivatives. 2. The Laplace transform of a function k(t) is defined as

Inverting the exponential Radon transform has a potential use for SPECT (single photon emission computed tomography) imaging in cases where a uniform attenuation can be approximated, such as in brain and abdominal imaging. Tretiak and Metz derived in the frequency domain an explicit inversion formula for the exponential Radon transform in two dimensions for parallel-beam collimator geometry. Progress has been made to extend the inversion formula for fan-beam and varying focal-length fan-beam (VFF) collimator geometries. These previous fan-beam and VFF inversion formulas require a spatially variant filtering operation, which complicates the implementation and imposes a heavy computing burden. In this paper, we present an explicit inversion formula, in which a spatially invariant filter is involved. The formula is derived and implemented in the spatial domain for VFF geometry (where parallel-beam and fan-beam geometries are two special cases). Phantom simulations mimicking SPECT studies demonstrate its accuracy in reconstructing the phantom images and efficiency in computation for the considered collimator geometries.