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This paper is devoted to the mathematical analysis of a spatially homogeneous thermostatted kinetic theory framework with an unbounded activity domain. The framework consists of a partial integro-differential equation with quadratic nonlinearity where the domain of the activity variable is the whole real line. Specifically the mathematical analysis refers firstly to the existence and uniqueness of the solution for the related initial boundary value problem; Secondly the investigations are addressed to the existence of a class of self-similar solutions by employing the Fourier transform method. In particular the main result is obtained for a nonconstant interaction rate and a nonconstant force field. Conclusions and perspectives are discussed in the last section of the paper. [ABSTRACT FROM AUTHOR]

Side-channel analysis is a critical threat to cryptosystems on the Internet of Things and in relation to embedded devices, and appropriate side-channel countermeasure must be required for physical security. A combined countermeasure approach employing first-order masking and desynchronization simultaneously is a general and cost-efficient approach to counteracting side-channel analysis. With the development of side-channel countermeasures, there are plenty of advanced attacks introduced to defeat such countermeasures. At CARDIS 2013, Belgarric et al. first proposed time-frequency analysis, a promising attack regarding the complexity of computation and memory compared to other attacks, such as conventional second-order side-channel analysis after synchronization. Nevertheless, their time-frequency analysis seems to have lower performance than expected against some datasets protected by combined countermeasures. It is therefore required to study the factors that affect the performance of time-frequency analysis. In this paper, we investigate Belgarric et al.'s time-frequency analysis and conduct a mathematical analysis in regard to the preprocessing of frequency information for second-order side-channel analysis. Based on this analysis, we claim that zero-mean preprocessing enhances the performance of time-frequency analysis. We verify that our analysis is valid through experimental results from two datasets, which are different types of first-order masked Advanced Encryption Standard (AES) software implementations. The experimental results show that time-frequency analysis with zero-mean preprocessing seems to have an enhanced or complementary performance compared to the analysis without preprocessing. [ABSTRACT FROM AUTHOR]

In the finite field setting, we show that the restriction conjecture associated to any one of a large family of d = 2n + 1-dimensional quadratic surfaces implies the n + 1-dimensional Kakeya conjecture (Dvir's theorem). This includes the case of the paraboloid over finite fields in which -1 is a square. We are able to partially reverse this implication using the sharp Kakeya maximal operator estimates of Ellenberg, Oberlin and Tao to establish the first finite field restriction estimates beyond the Stein--Tomas exponent in this setting. [ABSTRACT FROM AUTHOR]

Coherence is one common metric for cross‐dependence in multichannel signals. However, standard coherence does not sufficiently model many biological signals with complex dependence structures such as cross‐oscillatory interactions between a low‐frequency component in one signal and a high‐frequency component in another. The notion of cross‐dependence between low‐ and high‐frequency components, as defined in classical harmonizable processes, is still inadequate because it assumes time invariance and thus cannot capture cross‐frequency interactions that evolve over time. We construct a novel framework for modeling and estimating these dependencies under the replicated time series setting. Under this framework, we establish the novel concept of evolutionary dual‐frequency coherence and develop time‐localized estimators based on dual‐frequency local periodograms. The proposed nonparametric estimation procedure does not suffer from model misspecification. It uses the localized fast Fourier transform and hence is able to handle massive data. When applied to electroencephalogram data recorded in a motor intention experiment, the proposed method uncovers new and interesting cross‐oscillatory interactions that have been overlooked by the standard approaches. [ABSTRACT FROM AUTHOR]

Abstract We study the mapping properties of the Bargmann transform B on L p spaces of the real line. It is well known that B maps L 2 (R) isometrically onto the Fock space F 2. When 2 < p ≤ ∞ , we show that B maps L p (R) boundedly into the Fock space F p and that the mapping is not onto. When 1 ≤ p < 2 , we show that B maps L p (R) boundedly into the Fock space F q , where 1 / p + 1 / q = 1 , and that B does not map L p (R) into F p. There is no reasonable way to define the Bargmann transform on L p (R) when 0 < p < 1. [ABSTRACT FROM AUTHOR]

Salem’s necessary conditions for a trigonometric series to be the Fourier series of an integrable function are generalized to the nonperiodic case for functions in the Wiener algebra. Applications of the obtained result are given. [ABSTRACT FROM AUTHOR]

This is an attempt for mathematical formulation and general analytical solution of the most generalized thermal bending problem in the Cartesian domain. The problem has been formulated in the context of non-homogeneous transient heat equation subjected to Robin's boundary conditions. The general solution of the generalized thermoelastic problem has been discussed for temperature change, displacements, thermal stresses, deflection, and deformation. The most important feature of this work is any special case of practical interest may be readily obtained by this most generalized mathematical formulation and its analytical solution. There are 729 such combinations of possible boundary conditions prescribed on parallelepiped shaped region in the Cartesian coordinate system. The key idea behind the solution of heat equation is to transform the original initial and boundary value problem into eigenvalue problem through the Strum-Liouville theory. The finite Fourier transform has been applied with respect to space variables by choosing suitable normalized kernels. The well-posedness of the problem has been discussed by the existence, uniqueness, and stability of series solutions obtained analytically. The convergence of infinite series solutions also been discussed. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Donoho–Stark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi-shifts of group-like projections. We also prove the Hirschman–Beckner uncertainty principle for compact quantum groups and discrete quantum groups. Furthermore, we show Hardy's uncertainty principle for locally compact quantum groups in terms of bi-shifts of group-like projections. [ABSTRACT FROM AUTHOR]

We give the solution of the Turán, Fejér, Delsarte, Logan, and Bohman extremal problems for the Fourier transform on the hyperboloid ℍ or Lobachevsky space. We apply the averaging function method over the sphere and the solution of these problems for the Jacobi transform on the half-line. [ABSTRACT FROM AUTHOR]

We study a recent result of Bourgain, Clozel and Kahane, a version of which states that a sufficiently nice function f : R → R that coincides with its Fourier transform and vanishes at the origin has a root in the interval ( c , ∞ ) , where the optimal c satisfies 0.41 ≤ c ≤ 0.64 . A similar result holds in higher dimensions. We improve the one-dimensional result to 0.45 ≤ c ≤ 0.594 , and the lower bound in higher dimensions. We also prove that extremizers exist, and have infinitely many double roots. With this purpose in mind, we establish a new structure statement about Hermite polynomials which relates their pointwise evaluation to linear flows on the torus, and applies to other families of orthogonal polynomials as well. [ABSTRACT FROM AUTHOR]

We consider two problems on sections of convex bodies in hyperbolic space. The first one is a modified version of the Busemann–Petty problem. We look at conditions that guarantee a positive answer to this problem in all dimensions. The second problem is an analogue of a result of Makai, Martini, and Ódor about origin-symmetry. If in every direction the parallel section function has a critical value at zero, then the body is origin-symmetric. For both problems we use Fourier transform techniques. [ABSTRACT FROM AUTHOR]

General necessary conditions on the real parameters α, β, C, D for the function where ρ is the norm on R, to be positive definite on R, are obtained. For ρ( x) = | x|, a criterion on these parameters is obtained in the following cases: (i) p = 1or p = 2; (ii) 3 < p ≤ ∞ and n = 2. [ABSTRACT FROM AUTHOR]

This paper reports a new simulation technique for an aeroelastic system which responds to random external forces. Since the aeroelastic system including the effects of unsteady aerodynamics is ordinarily described in the frequency domain, the Inverse Discrete Fourier Transform (IDFT) can be utilized to simulate its random response. The response caused by the external random noise is calculated through a transfer function first in the frequency domain and then converted to the time domain. The objective of the present study is to provide mathematical time history data for evaluating the various estimation methods of the flutter boundary from subcritical responses in flight and/or wind tunnel testing. An example application to the method of flutter prediction is shown. The technique can also be used to evaluate the effects of the active control device coping with atmospheric turbulence. [ABSTRACT FROM PUBLISHER]

We introduce and study the notions of translation bounded tempered distributions, and autocorrelation for a tempered distribution. We further introduce the spaces of weakly, strongly and null weakly almost periodic tempered distributions and show that for weakly almost periodic tempered distributions the Eberlein decomposition holds. For translation bounded measures all these notions coincide with the classical ones. We show that tempered distributions with measure Fourier transform are weakly almost periodic and that for this class, the Eberlein decomposition is exactly the Fourier dual of the Lebesgue decomposition, with the Fourier-Bohr coefficients specifying the pure point part of the Fourier transform. We complete the project by looking at few interesting examples. [ABSTRACT FROM AUTHOR]

An essential generalization of the Lebedev index transform with the square of the Macdonald function is investigated. Namely, we consider a family of integral operators with the positive kernel | K ( i τ + α ) / 2 ( x ) | 2 , α ∈ R , x > 0 , τ ∈ R , where K μ ( z ) is the Macdonald function and i is the imaginary unit. Mapping properties such as the boundedness, compactness, invertibility are investigated for these operators and their adjoints in weighted L p spaces. Inversion theorems are proved. Important particular cases are exhibited. As an interesting application, a solution of the initial value problem for the second order differential difference equation, involving the Laplacian, is obtained. [ABSTRACT FROM AUTHOR]

Kahane provided a proof of a conjecture of Bateman and Diamond on Beurling generalized primes. The conjecture says that the L2-condition implies the PNT, i.e. π(x) ~ x/ x. Recently, Schlage-Puchta and Vindas showed that Beurling's condition in the form of a Cesàro mean with γ > 3/2 implies the PNT as well. Both are extensions of the Beurling PNT. We now show that the condition implies the PNT and that with k ≥ 0 implies the PNT also, where These results cover the preceding forms and extend further the Beurling PNT. As part of our argument, we extend first the theorems on Chebyshev bounds with correspondingly weaker conditions. These bounds play an important role in later proofs. [ABSTRACT FROM AUTHOR]

Luminescence decay functions describe the time dependence of the intensity of radiation emitted by electronically excited species. Decay phasor plots (plots of the Fourier sine transform vs. the Fourier cosine transform, for one or several angular frequencies) are being increasingly used in fluorescence, namely in lifetime imaging microscopy. In this work it is shown that the universal semicircle, locus of all exponential decay functions, can be viewed as the weighted sum of two spiraling phasors, one corresponding to a truncated exponential and the other to a shifted exponential. The geometric details of this recomposition are discussed. With area normalization, the decay functions form a subset in the universe of one-sided probability density functions, the same being valid for the phasor plots, which are parametric plots of the respective characteristic functions. [ABSTRACT FROM AUTHOR]

We use real Paley–Wiener theorems to give a characterization of the functions whose Fourier transform has support in antipodal points, and apply the results to study Roe's theorem. [ABSTRACT FROM AUTHOR]

The classical 픉ab transform was proclaimed as an extension to Fourier and Hartley transformations. In [17], authors have extended the transform to a space of generalized functions and they further have obtained some desired properties. This article discusses the 픉ab transform on certain space of generalized functions. Two spaces of Boehmians have been established and the extended 픉ab transform is also defined. Some general properties of the transform are given in some detail. [ABSTRACT FROM AUTHOR]

We improve the range of exponents for the restriction problem for the 3-d paraboloid over finite fields. The key new ingredient is a variant of the Bourgain–Katz–Tao finite field incidence theorem derived from sum-product estimates. In prime order fields, we give an explicit relationship between the exponent in this incidence theorem and restriction estimates for the paraboloid. [ABSTRACT FROM AUTHOR]

In this paper, we present criteria for the completeness of root subspaces of convolution operators on a finite interval with kernels equal to the Fourier transform of the ratio of two entire functions of exponential type from special classes. We describe a class of entire functions with a regular (in a certain sense) distribution of roots. [ABSTRACT FROM AUTHOR]

For the classes L(ℝ), r ∈ ℤ, we establish the upper and lower bounds for the quantities where µ, r ∈ ℤ, µ ≤ r, k ∈ ℕ, 0 < p ≤ 2, 0 < σ < ∞, 0 < t ≤ π/σ, and ψ is a nonnegative, measurable function summable on the closed interval [0, t] and not equivalent to zero. In the cases χ, where μ ∈ ℕ, 1/μ ≤ p ≤ 2, and χ(1, t), where 0 < t ≤ π/(2 σ), we obtain the exact values of these quantities. We also obtain the exact values of the average ν-widths of classes of functions defined in terms of the modulus of continuity ω* and the majorant ψ. [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]

We continue the description of the C*-algebras of low-dimensional nilpotent Lie groups as algebras of operator fields defined over their spectra. We show here that the C*-algebras of the three 6-dimensional nilpotent Lie groups having the group G5,2 as quotient group modulo the centre have norm controlled dual limits. [ABSTRACT FROM AUTHOR]

A new method for calculating the Fourier transform of a function is developed. This method is based on the generalization of the jump formula which is used in calculating the Fourier series of periodic functions. The developed formula is a very user-friend and handy equation. Besides the application of developed routine for calculating the Fourier transform, it is very useful in evaluating the convergence rate of the Fourier transform of the functions. It is believed that the proposed relation will very soon be appeared in Engineering Mathematics text books and communities. [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]

We define convolution of two distributions in the spaces, in two ways, which turn out to be equivalent, and show that the space is closed under convolution. We also define the Fourier transform of elements of, and prove the Parseval equality. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate a specific wave equation which generates the pre-Laguerre system, and prove that the pre-Laguerre system is related to the solution of the partial differential equation with some initial conditions. [Copyright &y& Elsevier]

Abstract: We obtain explicit estimates showing how the global norm of a band-limited function blows up, as it locally approximates a function with a jump in a derivative. As an application, we obtain bounds on how well a function, with a jump in a derivative, can be essentially time- and band-limited. [Copyright &y& Elsevier]

Abstract: We show that, under adequate norms, the Fourier transform is an isometry over a chain of nested weighted Sobolev spaces. As a result, an infinite number of useful Plancherel-like identities are derived. Possible extensions are discussed, giving rise to some open questions. [Copyright &y& Elsevier]

We construct smooth nonseparable compactly supported refinable functions that generate multiresolution analyses on L²(Rd), d > 1. Using these refinable functions we construct smooth nonseparable compactly supported orthonormal wavelet systems. These systems are nonseparable, in the sense that none of its constituent functions can be expressed as the product of two functions defined on lower dimensions. Both the refinable functions and the wavelets can be made as smooth as desired. Estimates for the supports of these scaling functions and wavelets, are given. [ABSTRACT FROM AUTHOR]

The harmonic ratio (HR), derived from the Fourier analysis of trunk accelerations, has been described in various ways as a measure of walking smoothness, walking rhythmicity, or dynamic stability. There is an increasing interest in applying the HR technique to investigate the impact of various pathologies on locomotion; however, explanation of the method has been limited. The aim here is to present a clear description of the mathematical basis of HRs and an understanding of their interpretation. We present harmonic theory, the interpretation of the HR using sinusoidal signals, and an example using actual trunk accelerations and harmonic analyses during limb-loading conditions. We suggest that the HR method may be better defined, not as a measure of rhythmicity or stability, but as a measure of step-to-step symmetry within a stride. [ABSTRACT FROM AUTHOR]

Solutions of the Navier-Stokes and Euler equations with initial conditions for 2D and 3D cases were obtained in the form of converging series, by an analytical iterative method using Fourier and Laplace transforms in [28, 29]. There the solutions are infinitely differentiable functions, and for several combinations of parameters numerical results are presented. This article provides a detailed proof of the existence, uniqueness and smoothness of the solution of the Cauchy problem for the 3D Navier-Stokes equations with any smooth initial velocity. When the viscosity tends to zero, this proof applies also to the Euler equations. [ABSTRACT FROM AUTHOR]

Certain relations between the Fourier transform of a function and the Hilbert transform of its derivative are revealed. They concern the integrability/non-integrability of both transforms. Certain applications are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we use some Fourier analysis techniques to find an exact solution to the Cauchy problem for the n-dimensional biwave equation in the upper half-space ℝ n × [0, +∞). [ABSTRACT FROM PUBLISHER]

Abstract: We solve a problem about the orthogonal complement of the space spanned by restricted shifts of functions in posed by M. Carlsson and C. Sundberg. [Copyright &y& Elsevier]

We study the weak type inequalities for the operator I - Fs Fc, where Fc and Fs are the cosine and sine Fourier transforms on the positive half line, respectively, and I is the identity operator. We also derive sharp constants in related weak type estimates for I - H픛, I - Hℝ and I - Hℝ+, where H픛, Hℝ and Hℝ+ denote the Hilbert transforms on the circle, on the real line and the positive half-line, respectively. Our main tool is the weak type inequality for orthogonal martingales, which is of independent interest. [ABSTRACT FROM AUTHOR]

Abstract: We give necessary and sufficient conditions on the Fourier transform of a generator function of a principal shift-invariant subspace of providing density order . Our starting point is a paper by de Boor, DeVore, and Ron [C. de Boor, R.A. DeVore, A. Ron, Approximation from shift-invariant subspaces of , Trans. Amer. Math. Soc. 341 (2) (1994) 787–806] and the new conditions that we present here involve the classical notion of the point of approximate continuity. In addition, we study properties of the approximation of a shift-invariant subspace of when it is dilated by a diagonalizable expansive linear map . Indeed, we present a necessary and sufficient condition on a principal shift-invariant subspace such that its union with itself dilated by integer powers of is dense in . [Copyright &y& Elsevier]

Abstract: Band-limited signals of finite energy, i.e., functions in , form the setting for much of signal processing. However, many signals of practical interest do not have finite energy. In this work we extend some results involving prolate spheroidal wave functions (PSWFs) to such signals. We consider those functions whose Fourier transforms have compact support in the sense of generalized functions and obtain convergence results for their PSWF expansions. [Copyright &y& Elsevier]

Abstract: In this paper, we study the optimal decay rates of solutions for the generalized Benjamin–Bona–Mahony equation in multi-dimensional space (). By using Fourier transform and the energy method, we obtain the convergence rates of the solutions under the condition that the initial data is small. The optimal decay rates obtained in this paper are found to be the same as the decay rate for the Heat equation. [Copyright &y& Elsevier]

The definition, properties and applications of the Sumudu transform to ordinary differential equations are described in [1-3]. In this manuscript we derive the formulae for the Sumudu and double Sumudu transforms of ordinary and partial fractional derivatives and apply them in solving Caputo-fractional differential equations. Our purpose here is to show the applicability of this new transform and its efficiency in solving such problems. [ABSTRACT FROM AUTHOR]

Abstract: We show that the de Branges theory provides a useful generalization of the Fourier transform (FT). The formulation is quite rich in that by selecting the appropriate parametrization, one can obtain spectral representation for a number of important cases. We demonstrate two such cases in this paper: the finite sum of elementary chirp-like signals, and a decaying chirp using Bessel functions. We show that when defined in the framework of de Branges spaces, these cases admit a representation very much similar to the spectral representation of a finite sum of sinusoids for the usual FT. [Copyright &y& Elsevier]

Non-harmonic Fourier transform is useful for the analysis of transient signals, where the integral kernel is from the boundary value of Möbius transform. In this note, we study the Paley-Wiener type extension theorems for the non-harmonic Fourier transform. Two extension theorems are established by using real variable techniques. [ABSTRACT FROM AUTHOR]

Abstract: In this article a technique for constructing p-ary bent functions from near-bent functions is presented. This technique is then used to obtain both weakly regular and non-weakly regular bent functions. In particular we present the first known infinite class of non-weakly regular bent functions. [Copyright &y& Elsevier]

The definition, properties and applications of the Sumudu transform to ordinary differential equations are described in [1-3]. In this manuscript we derive the formulae for the Sumudu and double Sumudu transforms of ordinary and partial fractional derivatives and apply them in solving Caputo-fractional differential equations. Our purpose here is to show the applicability of this new transform and its efficiency in solving such problems. [ABSTRACT FROM AUTHOR]

Abstract: A space of pseudoquotients is introduced that is shown to be isomorphic to the space of tempered distributions on . The Fourier transform is defined as a map from the space of pseudoquotients to the space of tempered distributions and as a transformation on pseudoquotients. [Copyright &y& Elsevier]

Abstract: For functions with Fourier transforms in we give necessary and sufficient conditions for f to belong to the generalized Lipschitz classes and in terms of behavior of . [Copyright &y& Elsevier]

The Fourier exchange theorem is generalized to 𝒮′-convolvable kernels, i.e. temperate distributions K(x, y) ∈ 𝒮′(ℝ2n ) for which K(x − y, y) is semitemperate with respect to x and partially summable with respect to y. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we show that there is no local equatorial characterization of bodies that embed in in odd dimensions for all p not even, . However, bodies that embed in for p odd are local equatorially characterizable provided that the dimension is even but not locally characterizable in general. This extends results given in Panina (1988) , Goodey and Weil (1992) , and Nazarov, Ryabogin and Zvavitch (2008) concerning the local equatorial characterization of zonoids and intersection bodies. These results were for bodies that embed in and respectively. [Copyright &y& Elsevier]