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In a recent breakthrough, Teyssier (Ann Probab 48(5):2323–2343, 2020) introduced a new method for approximating the distance from equilibrium of a random walk on a group. He used it to study the limit profile for the random transpositions card shuffle. His techniques were restricted to conjugacy-invariant random walks on groups; we derive similar approximation lemmas for random walks on homogeneous spaces and for general reversible Markov chains. We illustrate applications of these lemmas to some famous problems: the k-cycle shuffle, sharpening results of Hough (Probab Theory Relat Fields 165(1–2):447–482, 2016) and Berestycki, Schramm and Zeitouni (Ann Probab 39(5):1815–1843, 2011), the Ehrenfest urn diffusion with many urns, sharpening results of Ceccherini-Silberstein, Scarabotti and Tolli (J Math Sci 141(2):1182–1229, 2007), a Gibbs sampler, which is a fundamental tool in statistical physics, with Binomial prior and hypergeometric posterior, sharpening results of Diaconis, Khare and Saloff-Coste (Stat Sci 23(2):151–178, 2008). [ABSTRACT FROM AUTHOR]

Let d  ≥ 1. For any A ∈ Z d × d such that | det A | = 2 , we construct two families of Parseval wavelet frames with two generators. These generators have compact support, any desired number of vanishing moments, and any given degree of regularity. The first family is real valued while the second family is complex valued. To construct these families we use Daubechies low pass filters to obtain refinable functions, and adapt methods employed by Chui and He and Petukhov for dyadic dilations to this more general case. We also construct several families of Parseval wavelet frames with three generators having various symmetry properties. Our constructions are based on the same refinable functions and on techniques developed by Han and Mo and by Dong and Shen for the univariate case with dyadic dilations. [ABSTRACT FROM AUTHOR]

Asymptotic formulas are obtained for a class of integrals that are Fourier transforms of rapidly oscillating functions. These formulas contain special functions and generalize the well-known method of stationary phase. [ABSTRACT FROM AUTHOR]

A function f with bounded variation and integrable Hilbert transform of d f is locally absolutely continuous. We give a direct real-valued proof of this result and connect it with integrability problems for the Fourier transform of a function with bounded variation, in particular, with a Hardy-Littlewood theorem. [ABSTRACT FROM AUTHOR]

It is shown that the Laplace transform of an L (1 < p ≤ 2) function defined on the positive semiaxis satisfies the Hausdorff-Young type inequality with a positive weight in the right complex half-plane if and only if the weight is a Carleson measure. In addition, Carleson's weighted L inequality for the harmonic extension is given with a numeric constant. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to formulate and prove an L- L analog of Miyachi's theorem for connected nilpotent Lie groups with noncompact center for 2 ≤ p, q ≤ +∞. This allows us to solve the sharpness problem in both Hardy's and Cowling-Price's uncertainty principles. When G is of compact center, we show that the aforementioned uncertainty principles fail to hold. Our results extend those of [1], where G is further assumed to be simply connected, p = 2, and q = +∞. When G is more generally exponential solvable, such a principle also holds provided that the center of G is not trivial. Representation theory and a localized Plancherel formula play an important role in the proofs. [ABSTRACT FROM AUTHOR]

We give simple proofs of the essential properties of the Beurling-Selberg function and its odd part by viewing them as solutions to difference equations. [ABSTRACT FROM AUTHOR]

Nature of the Natural transform is, it converges to Laplace and Sumudu transform. The theme of this paper is to give detailed study of Natural transform. The Natural transform is derived from the Fourier Integral. We showed it to the theoretical dual of Laplace and Sumudu transforms. This work includes Natural-multiple shift theorems, Bromwich contour integral and Heviside's Expansion formula for Inverse Natural transform. [ABSTRACT FROM AUTHOR]

A k-dimensional hypertree X is a k-dimensional complex on n vertices with a full ( k−1)-dimensional skeleton and $\binom{n-1}{k}$ facets such that H k( X;ℚ)=0. Here we introduce the following family of simplicial complexes. Let n, k be integers with k+1 and n relatively prime, and let A be a ( k+1)-element subset of the cyclic group ℤ n. The sum complex X A is the pure k-dimensional complex on the vertex set ℤ n whose facets are σ⊂ℤ n such that | σ|= k+1 and ∑ x∈ σ x∈ A. It is shown that if n is prime, then the complex X A is a k-hypertree for every choice of A. On the other hand, for n prime, X A is k-collapsible iff A is an arithmetic progression in ℤ n. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a locally compact group of type I and its dual space. Roughly speaking, qualitative uncertainty principles state that the concentration of a nonzero integrable function on G and of its operator-valued Fourier transform on is limited. Such principles have been established for locally compact abelian groups and for compact groups. In this paper we prove generalizations to the considerably larger class of groups with finite dimensional irreducible representations. [Copyright &y& Elsevier]

Consider the problem of finding two coefficients one of which is under the sign of the integral in the hyperbolic equation and represents the memory of a medium and the other determines the regular part of an impulse source. Additionally, the Fourier transform of the trace of the solution of the direct problem on the hyperplane y = 0 for two different values of the transformation parameter is given. We establish an estimate of the stability of the solution of the inverse problem under consideration and also the uniqueness theorem. [ABSTRACT FROM AUTHOR]

This paper investigates a multi-resolution digital Earth model called PYXIS, which was developed by PYXIS Innovation Inc. The PYXIS hexagonal grids employ an efficient hierarchical labeling scheme for addressing pixels. We provide a recursive definition of the PYXIS grids, a systematic approach to the labeling, an algorithm to add PYXIS labels, and a discussion of the discrete Fourier transform on PYXIS grids. [ABSTRACT FROM AUTHOR]

We use the Freiman theorem in arithmetic combinatorics to show that if the Fourier transform of certain measures satisfies sufficiently bad estimates, then the support of the measure possesses an additive structure. The result is then discussed in light of the Falconer distance problem. [ABSTRACT FROM AUTHOR]

This paper addresses the permissible problem of the product of two intrinsically stationary variograms and that of a power function of an intrinsically stationary variogram in space and time. It results in several new classes of space-time variograms, as well as power-law and other long-range dependent covariance models. In particular, the product of two isotropic variograms that are formulated in terms of Bernstein functions is shown to be a valid variogram, and the largest permissible power is determined for the power of an intrinsically stationary variogram that is constructed via the Bernstein function. [ABSTRACT FROM AUTHOR]

The wavelet theory is a powerful method for processing different physical and chemical signals. The fractional wavelet can be obtained from the a fractional window. In this brief report we present a new wavelet transform which is obtained from Agnesi window. This class of wavelet transform is a flexible one because dependents on two parameters (dilatation or contraction, position). It cover also combinations of discrete and continuous parameter discretisations (hybrid cases). [ABSTRACT FROM AUTHOR]

This paper introduces a general and axiomatic approach to linear signal processing (SP) that we refer to as the algebraic signal processing theory (ASP). Basic to ASP is the linear signal model defined as a triple (A, M, Φ) where familiar concepts like the filter space and the signal space are cast as an algebra A and a module M, respectively. The mapping Φ generalizes the concept of a z-transform to bijective linear mappings from a vector space of signal samples into the module M. Common concepts like filtering, spectrum, or Fourier transform have their equivalent counterparts in ASP. Once these concepts and their properties are defined and understood in the context of ASP, they remain true and apply to specific instantiations of the ASP signal model. For example, to develop signal processing theories for infinite and finite discrete time signals, for infinite or finite discrete space signals, or for multidimensional signals, we need only to instantiate the signal model to one that makes sense for that specific class of signals. Filtering, spectrum, Fourier transform, and other notions follow then from the corresponding ASP concepts. Similarly, common assumptions in SP translate into requirements on the ASP signal model. For example, shift-invariance is equivalent to A being commutative. For finite (duration) signals shift invariance then restricts A to polynomial algebras. We explain how to design signal models from the specification of a special filter, the shift. The paper illustrates the general ASP theory with the standard time shift, presenting a unique signal model for infinite time and several signal models for finite time. The latter models illustrate the role played by boundary conditions and recover the discrete Fourier transform (DFT) and its variants as associated Fourier transforms. Finally, ASP provides a systematic methodology to derive fast algorithms for linear transforms. This topic and the application of ASP to space dependent signals and to multidimensional signals are pursued in companion papers. [ABSTRACT FROM AUTHOR]

A class of compactly supported Schrödinger potentials in dimension two is given for which the inverse scattering method related to the Novikov-Veselov evolution equation is well-defined. There is no smallness assumption on the initial potential. Regularity results are proven for the direct and inverse scattering transforms, also called nonlinear Fourier transforms. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the thermoelastic system with second sound in one space variable. The mathematical model is a hyperbolic partial differential system with damping term which brings about some parabolic properties. The pointwise estimates of the solution are obtained and from which the generalized Huygens'' principle is shown. Our approach is based on detailed analysis on the Green function of the system. [Copyright &y& Elsevier]

The basic principles of a new version of Fourier transformation is presented. This new version was applied to solve some main problems such as smoothing, and denoising in gamma spectroscopy. The mathematical procedures were first tested by simulated data and then by actual experimental data. [ABSTRACT FROM AUTHOR]

We prove an analog of the classical Titchmarsh theorem on the image under the Fourier transform of a set of functions satisfying the Lipschitz condition in L 2 for functions on noncompact rank 1 Riemannian symmetric spaces. [ABSTRACT FROM AUTHOR]

Abstract: Spatial statistics is one of the major methodologies of image analysis, field trials, remote sensing, and environmental statistics. The standard methodology in spatial statistics is essentially based on the assumption of stationary and isotropic random fields. Such assumptions might not hold in large heterogeneous fields. Thus, it is important to understand when stationarity and isotropy are reasonable assumptions. Most of the work that has been done so far to test the nonstationarity of a random process is in one dimension. Unfortunately, there is not much literature of formal procedures to test for stationarity of spatial stochastic processes. In this manuscript, we consider the problem of testing a given spatial process for stationarity and isotropy. The approach is based on a spatial spectral analysis, this means spectral functions which are space dependent. The proposed method consists essentially in testing the homogeneity of a set of spatial spectra evaluated at different locations. In addition to testing stationarity and isotropy, the analysis provides also a method for testing whether the observed process fits a uniformly modulated model, and a test for randomness (white noise). Applications include modeling and testing for nonstationary of air pollution concentrations over different geo-political boundaries. [Copyright &y& Elsevier]

The difference or a linear combination of two space- time covariance functions (or variograms is not necessarily a valid covariance function (or variogram). In general, there seems no simple condition that makes a linear combination permissible, unless its coefficients are non-negative. The permissibility is investigated in this paper for the linear combination of two spatial or spatio-temporal covariance functions (or variograms) isotropic in space so that we obtain flexible classes of spatial or spatio-temporal covariance functions with various properties such as long-range dependence and having different signs. [ABSTRACT FROM AUTHOR]

In Brackx et al., 2004 (F. Brackx, R. Delanghe and F. Sommen (2004). Spherical means and distributions in Clifford analysis. In: Tao Qian, Thomas Hempfling, Alan McIntosch and Frank Sommen (Eds.), Advances in Analysis and Geometry: New Developments Using Clifford Algebra. Trends in Mathematics, pp. 65–96. Birkhäuser, Basel.), some fundamental higher dimensional distributions have been reconsidered within the framework of Clifford analysis. Here, the Fourier transforms of these distributions are calculated, revealing a.o. the Fourier symbols of some important translation invariant (convolution) operators, which can be interpreted as members of the considered families. Moreover, these results are the incentive for calculating the Fourier symbols of some differential operators which are at the heart of Clifford analysis, but do not show the property of translation invariance and hence, can no longer be interpreted as convolution operators. [ABSTRACT FROM AUTHOR]

We consider an exterior problem without initial conditions for a class of equations of parabolic type. An existence and uniqueness theorem for the solution of this problem is proved. In the proof, Hardy's inequality for function spaces with derivatives of nonintegral order (a result obtained earlier by the author) is essentially used. [ABSTRACT FROM AUTHOR]

Mathematical diffraction theory is concerned with the Fourier analysis of the autocorrelations of translation bounded measures onσ-compact locally compact Abelian groups, withn-dimensional Euclidean space being the most important example. This mathematical approach provides new insight into the theory of kinematic diffraction, in particular into the distinction of pure Bragg versus diffuse scattering inX-ray diffraction. A number of unexpected properties can then be formulated and dealt with in a rigorous way. [ABSTRACT FROM AUTHOR]

An efficient algorithm exists for learning disjunctive normal form (DNF) expressions in the uniform-distribution PAC learning model with membership queries (J. Comput. System Sci. 55 (1997) 414), but in practice the algorithm can only be applied to small problems. We present several modifications to the algorithm that substantially improve its asymptotic efficiency. First, we show how to significantly improve the time and sample complexity of a key subprogram, resulting in similar improvements in the bounds on the overall DNF algorithm. We also apply known methods to convert the resulting algorithm to an attribute efficient algorithm. Furthermore, we develop a technique for lower bounding the sample size required for PAC learning with membership queries under a fixed distribution and apply this technique to produce a lower bound on the number of membership queries needed for the uniform-distribution DNF learning problem. Finally, we present a learning algorithm for DNF that is attribute efficient in its use of random bits. [Copyright &y& Elsevier]

Some classical real inversion formulas, such as those concerning Fourier, Laplace and Stieltjes transforms, are unified in a way which allows us to give rates of convergence. As illustration, the case of the Fourier transform is considered. [ABSTRACT FROM AUTHOR]

This paper is primarily concerned with proving the Lp boundedness of Marcinkiewicz integral operators with kernels belonging to certain block spaces. We also show the optimality of our condition on the kernel for the L2 boundedness of the Marcinkiewicz integral. [Copyright &y& Elsevier]