Inverse Measures, the Inversion Formula, and Discontinuous 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 ∞.

Inversion Formula for Continuous Multifractals

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.

Quaternionic one-dimensional linear canonical transform

The quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) is a nontrivial generalization of the quaternion Wigner-Ville distribution to the linear canonical transform (LCT) domain. In the present paper, we establish a fundamental relationship between the QWVD-LCT and the quaternion Fourier transform (QFT). Based on this fact, we provide alternative proof of the well-known properties of the QWVD-LCT such as inversion formula and Moyal formula. We also discuss in detail the relationship among the QWVD-LCT and other generalized transforms. Finally, based on the basic relation between the quaternion ambiguity function associated with the linear canonical transform (QAF-LCT) and the QFT, we present some important properties of the QAF-LCT.

A new integral transform and its applications

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

In a previous article we proposed an algebraic setting in which to perform harmonic analysis on noncompact, nondiscrete quantum groups and in particular, on quantum E(2). In the present paper we shall explicitly construct Fourier transforms between quantum E(2) and its Pontryagin dual, involving Hahn—Exton q-Bessel functions as kernel, prove Plancherel and inversion formulas, etc. We also develop a theory of q-Hankel transformation of entire functions, based on the definition proposed by Koornwinder and Swarttouw.

The present paper is devoted to study the transform by the index of the Legendre function which is known as the Mehler‐Fock transform. Mapping properties of the Mehler‐Fodr transform in the weighted space Lp(ω(t); IR+) are given as the inversion formula. The image space is also characterized.

In an upcoming article we study harmonic analysis on the quantum E(2) group within an algebraic framework: we explicitly construct Fourier transforms between quantum E(2) and its Pontryagin dual, involving q-Bessel functions as kernel, prove Plancherel & inversion formulas etc. In the present paper we propose an algebraic setting in which to perform harmonic analysis on non-compact, non-discrete quantum groups and in particular on quantum E(2). We are mainly concerned with the construction of positive and faithful invariant functionals on an algebraic level, KMS properties, etc.

In an upcoming article we study harmonic analysis on the quantum E(2) group within an algebraic framework: we explicitly construct Fourier transforms between quantum E(2) and its Pontryagin dual, involving q-Bessel functions as kernel, prove Plancherel & inversion formulas etc. In the present paper we propose an algebraic setting in which to perform harmonic analysis on non-compact, non-discrete quantum groups and in particular on quantum E(2). We are mainly concerned with the construction of positive and faithful invariant functionals on an algebraic level, KMS properties, etc.

The aim of this paper is to develop a new harmonic analysis related to a Bessel type operator on the real line: We define the canonical Fourier Bessel transform and study some of its important properties. We prove a Riemann–Lebesgue lemma, inversion formula and operational formulas for this transformation. We derive Plancherel theorem and Babenko inequality for In the present paper, several uncertainty inequalities and theorems for the canonical Fourier Bessel transform are given, including the Heisenberg inequality, Hardy theorem, Nash-type inequality, Carlson-type inequality, global uncertainty principle, local uncertainty principle, logarithmic uncertainty principle in terms of entropy and Miyachi uncertainty principle.

In this paper an alternative derivation of Katsevich's cone-beam image reconstruction algorithm is presented. The starting point is the classical Tuy's inversion formula. After (i) using the hidden symmetries of the intermediate functions, (ii) handling the redundant data by weighting them, (iii) changing the weighted average into an integral over the source trajectory parameter, and (iv) imposing an additional constraint on the weighting function, a filtered backprojection reconstruction formula from cone beam projections is derived. The following features are emphasized in the present paper: First, the nontangential condition in Tuy's original data sufficiency conditions has been relaxed. Second, a practical regularization scheme to handle the singularity is proposed. Third, the derivation in the cone beam case is in the same fashion as that in the fan-beam case. Our final cone-beam reconstruction formula is the same as the one discovered by Katsevich in his most recent paper. However, the data sufficiency conditions and the regularization scheme of singularities are different. A detailed comparison between these two methods is presented.

In previous work an “inversion formula” has been derived which expresses a function of two variables (such as the distribution of x-ray attenuation coefficients in a cross-section) in terms of its integrals along lines diverging from a point which moves around a circle. This inversion formula provides the same kindof theoretical background to divergent beam data collection that Radon’s theorem provides for parallel beam data collection. This formula is important, since rapid data collection in tomographic devices currently implies divergent beam data collection. For practical implementation the formula needs to be “regularized”.In the present paper a theorem is presented and is used for showing the appropriateness of a wide class of functions for the regularization of the inversion formula for divergent beams. Each of these functions is based on a filter and leads to an efficient convolution type numerical implementation. A number of filters are investigated in detail, and their behavior in reconstructing...

Let $$\mu $$ be a self-similar measure generated by an IFS $$\varPhi =\{\phi _i\}_{i=1}^\ell $$ of similarities on $${{\mathbb {R}}}^d$$ ( $$d\ge 1$$ ). When $$\varPhi $$ is dimensional regular (see Definition 1.1), we give an explicit formula for the $$L^q$$ -spectrum $$\tau _\mu (q)$$ of $$\mu $$ over [0, 1], and show that $$\tau _\mu $$ is differentiable over (0, 1] and the multifractal formalism holds for $$\mu $$ at any $$\alpha \in [\tau _\mu '(1),\tau _\mu '(0+)]$$ . We also verify the validity of the multifractal formalism of $$\mu $$ over $$[\tau _\mu '(\infty ),\tau _\mu '(0+)]$$ for two new classes of overlapping algebraic IFSs by showing that the asymptotically weak separation condition holds. For one of them, the proof appeals to the recent result of Shmerkin (Ann. Math. (2) 189(2):319–391, 2019) on the $$L^q$$ -spectrum of self-similar measures.