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Compared with conventional HVDC systems, the hybrid multiterminal HVDC (Hybrid-MTDC) system has a more sophisticated network topology composed of line commutated converter (LCC) and modular multilevel converter (MMC) stations, which leads to dc fault discrimination being difficult and ambiguous. Furthermore, for long transmission lines, high fault resistances, and different converter impedance characteristics, the reliable discrimination of external and internal faults located at the remote end and T areas is particularly demanding. Therefore, considering the frequency characteristics of dc boundaries and equivalent converter impedances and dc line models, this article constructs a time-domain convolution power. The performance of the convolution power in various fault scenarios is mathematically elaborated. Due to its good decomposition ability for short-time nonstationary signals, wavelet decomposition is utilized to extract the specific frequency bands (FBs) of traveling waves; thus, the convolution power in different FBs can be calculated. On this basis, a convolution power-based protection scheme is proposed for identifying external and internal faults, locating a dc bus fault, and selecting faulty poles. PSCAD/EMTDC simulation results show that various dc faults can be correctly identified within 3 ms, and the proposed scheme has high reliability and sensitivity, which is beneficial for the protection of Hybrid-MTDC systems.

We investigate the asymptotic behaviour of the difference between the tails of a self-decomposable distribution with a two-sided regularly varying density on the real line and its Levy measure. Moreover, we study the second-order asymptotic behaviour of the tail of the t-th convolution power of a self-decomposable distribution with a two-sided regularly varying density.

We characterize the second order subexponentiality of an infinitely divisible distribution on the real line under an exponential moment assumption. We investigate the asymptotic behaviour of the difference between the tails of an infinitely divisible distribution and its Lévy measure. Moreover, we study the second order asymptotic behaviour of the tail of the $t$th convolution power of an infinitely divisible distribution. The density version for a self-decomposable distribution on the real line without an exponential moment assumption is also given. Finally, the regularly varying case for a self-decomposable distribution on the half line is discussed.

We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some methods of this approximation for functions in a class induced by the convolution with a given function, and prove upper bounds of $L_p$-the approximation convergence rate by these methods, when $n \to \infty$, for $1 < p < \infty$, and lower bounds of the quantity of best approximation of this class by arbitrary linear combinations of $n$ translates of arbitrary function, for the particular case $p=2$.

ElGamal system was introduced by Taher ElGamal in his landmark paper in 1985 [1]. ElGamal is an important asymmetric cryptosystem that is based on the discrete logarithm problem. In brief, this cryptosystem relies on the difficulty of computing discrete logarithms over finite fields [1]. Towards solving discrete logarithms, Shor demonstrated the existence of a polynomial algorithm that can solve these logarithms on quantum machines [2]. In this paper we propose the Convolution Power Cryptosystem (CPC-H2); an extension to the ElGamal cryptosystem that is based on a novel hard problem. CPC-H2 is designed to defend against a Chosen Ciphertext Attack. We show that CPC-H2 is less complicated and has a higher security level than the Number Theory Research Unit (NTRU).

We study the following $$(q-1)$$ ( q - 1 ) th convex ordering relation for qth convolution power of the difference of probability distributions $$\mu $$ μ and $$\nu $$ ν $$\begin{aligned} (\nu -\mu )^{*q}\ge _{(q-1)cx} 0 , \quad q\ge 2, \end{aligned}$$ ( ν - μ ) ∗ q ≥ ( q - 1 ) c x 0 , q ≥ 2 , and we obtain the theorem providing a useful sufficient condition for its verification. We apply this theorem for various families of probability distributions and we obtain several inequalities related to the classical interpolation operators. In particular, taking binomial distributions, we obtain a new, very short proof of the inequality given recently by Abel and Leviatan (2020).

Let G be a locally compact abelian group and let M ( G ) be the convolution measure algebra of G. A measure μ ∈ M ( G ) is said to be power bounded if sup n ≥ 0 ⁡ ‖ μ n ‖ 1 ∞ , where μ n denotes nth convolution power of μ. We show that if μ ∈ M ( G ) is power bounded and A = [ a n , k ] n , k = 0 ∞ is a strongly regular matrix, then the limit lim n → ∞ ⁡ ∑ k = 0 ∞ a n , k μ k exists in the weak⁎ topology of M ( G ) and is equal to the idempotent measure θ, where θ ˆ = 1 int F μ . Here, θ ˆ is the Fourier-Stieltjes transform of θ, F μ : = { γ ∈ Γ : μ ˆ ( γ ) = 1 } , and 1 int F μ is the characteristic function of int F μ . Some applications are also given.

Large-scale discrete convolution, well-known to be computationally intensive, is a fundamental algorithmic building block in many computer vision and artificial intelligence applications. This work presents a novel stochastic-based hardware architecture and design that computes discrete convolution based on the widely-used convolution theorem. Our approach has three advantages. First, it can achieve approximately ${\mathrm O} (1)$ in algorithmic complexity for any given absolute error bound $d$ and any given input vector size $N$ . This computing complexity, when compared with ${\mathrm O} (N^2)$ and ${\mathrm O} (N \log N)$ operations for conventional multiplier-based and FFT-based architectures respectively, represents a significant improvement, although at the cost of degraded computing accuracy. Second, to achieve a given computing accuracy, our proposed stochastic-based convolution can be analytically proven to only require a moderate number of random samples, e.g., 788 random samples can achieve 95 percent accuracy at 99 percent confidence level for a convolution with $N=128$ . Third, this proposed stochastic-based architecture is highly fault-tolerant because the information to be processed is encoded with a large ensemble of random samples. As such, the local perturbations of its computing accuracy will be dissipated globally, thus becoming inconsequential to the final overall results. We believe that, being highly scalable and energy efficient, our stochastic-based convolution architecture is well-suited for many real-time embedded applications, especially those perception-based computing tasks that are inherently fault-tolerant. In short, this work provides an elegant way to tradeoff between computing accuracy and computing performance/hardware efficiency for many real-world convolution-based applications.

This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. LetGbe a compact group andHa closed subgroup ofG. Letμbe the normalizedG-invariant measure over the compact homogeneous spaceG/Hassociated with Weil's formula and. We then present a structured class of abstract linear representations of the Banach convolution function algebrasLp(G/H,μ).

We show that nontrivial convolution operators on certain spaces of entire functions on E are frequently hypercyclic when E is a normed space and when E is the strong dual of a Frechet nuclear space. We also obtain results of existence and approximation for convolution equations on certain spaces of entire functions on arbitrary locally convex spaces.

In this paper, we investigate majorization properties for certain classes of analytic functions defined by convolution structure. Also we point out some new and known consequences of our main result.

In this note we study the relation between oscillatory integral operators and convolution operators, and also the sharpness of L p L^p -estimates for singular oscillatory integral operators.

A certain convolution equation is studied on a large finite interval. This equation arose in acoustics for description of a wave conductor surface with a bed of ice. The main feature of this equation is that the symbol of the corresponding operator has zeros of power order in the dual variable, so that the inverse operator is a long-range one. A complete power-order asymptotic expansion is constructed for the kernel of the inverse operator as the length of the interval tends to infinity.

In this paper we define a more general convolution product of functionals on Wiener space and develop the fundamental relationships between the generalized Fourier–Feynman transform and the convolution product.

In this article we first introduce some results about the binomial, trinomial, and quadrinomial convolution sums of certain divisor functions and then we provide an identity for the multinomial convolution sums of the divisor function σ r ♯ ( n ) : = ∑ d | n n / d ≡ 1 ( 4 ) d r − ( − 1 ) r ∑ d | n n / d ≡ − 1 ( 4 ) d r .

The middle convolution, introduced by Katz and developed by Dettweiler–Reiter, Oshima and others, defines an operation of reduction of linear ordinary differential equations with polynomial coefficients. In this paper, employing an idea of the exact steepest descent method proposed by Aoki–Kawai–Takei, we study how to determine the complete Stokes geometry and how to obtain the Borel summability of WKB solutions of a higher order linear ordinary differential equation with a large parameter when it is reduced to a second order equation via middle convolution. To show the practical usefulness of the method, we also investigate some concrete examples numerically.

Suppose V{\nu} is the pseudo-variance function of the Cauchy-Stieltjes Kernel (CSK) family K+({\nu}) generated by a non degenerate probability measure {\nu} with support bounded from above. We determine the formula for pseudo-variance function (or variance function V{\nu} in case of existence) under boolean additive convolution power. This formulas is used to identify the relation between variance functions under Boolean Bercovici-Bata Bijection between probability measures. We also gives the connection between boolean cumulants and variance function and we relate boolean cumulants of some probability measures to Catalan numbers and Fuss Catalan numbers., Comment: We determine the formula for pseudo-variance function (or variance function V{\nu} in case of existence) under boolean additive convolution power

Two hypotheses on the class $\mathcal{L}(\gamma)$ in the class $\mathcal{OS}\cap \mathcal{ID}$ are discussed. Two weak hypotheses on the class $\mathcal{L}(\gamma)$ in the class $\mathcal{OS}\cap \mathcal{ID}$ are proved. A necessary and sufficient condition in order that, for every $t >0$, the $t$-th convolution power of a distribution in the class $\mathcal{OS}\cap \mathcal{ID}$ belongs to the class $\mathcal{L}(\gamma)$ is given., Comment: 19 pages

Let be a power function \(f_{r,M}(s)\) defined for every \(s\) within the finite set \(M\) as follows \(f_{r,M}(s)= \begin{cases} s^r, \ &s\in M,\\ 0, \ &\mathrm{otherwise}. \end{cases}\) Let a discrete convolution of \(f_{r,M}(s)\) be denoted as follows \(\mathrm{Conv}_{r,M}[n]=(f_{r,M}*f_{r,M})[n]\). Let a real coefficients \(A_{m,j}\) be given by the following recurrence \( A_{m,j} = \begin{cases} 0, & \mathrm{if } \ jm, \\ (2j+1)\binom{2j}{j} \sum_{d=2j+1}^{m} A_{m,d} \binom{d}{2j+1} \frac{(-1)^{d-1}}{d-j} B_{2d-2j}, & \mathrm{if } \ 0 \leq j < m, \\ (2j+1)\binom{2j}{j}, & \mathrm{if } \ j=m. \end{cases}\) In this paper we show that for every $n>0$ the following odd-power identities involving coefficients \(A_{m,j}\) and convolution transform \(\mathrm{Conv}_{r,M}[n]\) hold \(\begin{split} n^{2m+1}+1&=\sum_{r=0}^{m}A_{m,r}\mathrm{Conv}_{r,\mathbb{N}}[n],\\ n^{2m+1}-1&=\sum_{r=0}^{m}A_{m,r}\mathrm{Conv}_{r,\mathbb{Z}_{>0}}[n],\\ n^{2m+1}&=\sum_{r=0}^{m}A_{m,r}\sum_{k=1}^{n} k^r(n-k)^r\\ &=\sum_{r=0}^{m}A_{m,r}\sum_{k=0}^{n-1} k^r(n-k)^r. \end{split} \), 6 pages, {"references":["Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 10, 1972.","Knuth, D. E., Johann Faulhaber and Sums of Powers, pp. 9-10., arXiv preprint, arXiv:math/9207222v1 [math.CA], 1992.","Worpitzky, J.. \"Studien uber die Bernoullischen und Eulerschen Zahlen..\" Journal fur die reine und angewandte Mathematik 94 (1883): 203-232. http://eudml.org/doc/148532.","Graham, Ronald, Knuth Donald, and Oren Patashnik (1989-01-05). \"Special Numbers\". Concrete Mathematics (2nd ed.). Addison Wesley Longman Publishing Co. p. 262. ISBN 0-201-14236-8.","D. V. Widder et al, \"The Convolution Transform.\", Bull. Amer. Math. Soc. 60 (1954), 444-456. https://doi.org/10.1090/S0002-9904-1954-09828-2.","Weisstein, Eric W. \"Power.\" From MathWorld--A Wolfram Web Resource, equation 12.","Abramowitz, M. and Stegun, I. A. (Eds.). \"Multinomial Coefficients.\" §24.1.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 823-824, 1972."]}

We study solutions to convolution equations for functions with discrete support in ℝ n , a special case being functions with support in the integer points. The Fourier transform of a solution can be extended to a holomorphic function in some domains in ℂ n , and we determine possible domains in terms of the properties of the convolution operator.

In this study, we consider the use of the convolution method for constructing approximations comprising fuzzy number sequences with useful properties for a general fuzzy number. We show that this convolution method can generate differentiable approximations in finite steps for fuzzy numbers with finite non-differentiable points. In previous studies, this convolution method was only used for constructing differentiable approximations of continuous fuzzy numbers, the possible non-differentiable points of which were the two endpoints of the 1-cut. The construction of smoothers is a key step in the process for producing approximations. We also show that if appropriate smoothers are selected, then we can use the convolution method to provide approximations that are differentiable, Lipschitz, and that also preserve the core.

The concept of variance is invoked as a means to quantify robustly the spread of a radiation power pattern. A precise definition for variance is given in the context of Heisenberg’s uncertainty principle. A methodology is provided to quantify the variance exactly and in closed form using autocorrelation and convolution integrals. Several examples are provided to demonstrate both the power of the variance definition and the straightforwardness of the mathematical process. All mathematical results provided are new and unreported in the open literature. Additionally, a new Heisenberg product is formulated that is a relevant addition to the theory of line source radiation. Data obtained from numerical integration are provided to validate the closed-form analytical results derived herein.

We establish sharp estimates for the best approximations of convolution classes by entire functions of exponential type. To obtain these estimates, we propose a new method for testing Nikol’skiĭ-type conditions which is based on kernel periodization with an arbitrarily large period and ensuing passage to the limit. As particular cases, we obtain sharp estimates for approximation of convolution classes with variation diminishing kernels and generalized Bernoulli and Poisson kernels.

Some novel convolution properties for meromorphically multivalent functions associated with the generalized Bessel functions of the first kind are derived.

Our paper investigates the nonparametric estimation of cumulative distribution functions of nonnegative valued random variables using convolution power kernels. Our proposed consistent estimator avoids boundary effects near the origin. We present its asymptotic properties and give a short simulation study.

We study a representation for the inverse transform of the generalised Fourier–Feynman transform on the function space $C_{a,b}[0,T]$ which is induced by a generalised Brownian motion process. To do this, we define a transform via the concept of the convolution product of functionals on $C_{a,b}[0,T]$. We establish that the composition of these transforms acts like an inverse generalised Fourier–Feynman transform and that the transforms are vector space automorphisms of a vector space ${\mathcal{E}}(C_{a,b}[0,T])$. The vector space ${\mathcal{E}}(C_{a,b}[0,T])$ consists of exponential-type functionals on $C_{a,b}[0,T]$.

We first define the Hardy-Sobolev spaces associated with twisted convolution; then we give the atomic decomposition. As an application, we consider the endpoint version of the div-curl theorem for the twisted convolution.

We introduce a generalized convolution product $(F*G)_{\vec\alpha,\vec\beta}$ for integral transform ${\mathcal F}_{\gamma,\eta}$ for functionals defined on $K[0,T]$, the space of complex valued continuous functions on $[0,T]$ that vanish at zero. We study some interesting properties of our generalized convolution product and establish various relationships that exist among the generalized convolution product, the integral transform, and the first variation for functionals defined on $K[0,T]$. We also discuss the associativity of the generalized convolution product.

It is well known that the optimal transmission conditions (TCs) give (in theory) the best convergence for the Schwarz waveform relaxation (SWR) algorithm. These TCs are of convolution type and are not systematically explored yet, partly because numerical computation of convolutions is expensive. However, if the governing equation is also of convolution type, such as the Volterra integro-partial differential equations with weakly singular kernels studied in this paper, this kind of TCs seems a very natural choice. For a class of convolution TCs including the optimal one derived from the space-time continuous analysis, we perform a convergence analysis for the SWR algorithm at the fully discretized level in the framework of convolution quadratures, which permits us to precisely capture the effects of the discretizations on the convergence rate. By optimizing the parameter in the TCs, we show that the convergence factor can be reduced to a value around 0.0730 if an A-stable convolution quadrature is used. Th...

We consider a continuous, infinitely divisible random field in ℝd, d = 1, 2, 3, given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. For a large class of such random fields, we compute the asymptotic probability that the excursion set at level x contains some rotation of an object with fixed radius as x → ∞. Our main result is that the asymptotic probability is equivalent to the right tail of the underlying Lévy measure.

Fractional cosine transform (FRCT) and fractional sine transform (FRST), which are closely related to the fractional Fourier transform (FRFT), are useful mathematical and optical tool for signal processing. Many properties for these transforms are well investigated, but the convolution theorems are still to be determined. In this paper, we derive convolution theorems for the fractional cosine transform (FRCT) and fractional sine transform (FRST) based on the four novel convolution operations. And then, a potential application for these two transforms on designing multiplicative filter is presented. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper we study the approximation of stable linear time-invariant systems for the Paley–Wiener space \(\mathcal {PW}_{\pi }^2\), i.e., the set of bandlimited functions with finite \(L^2\)-norm, by convolution sums. It is possible to use either, the convolution sum where the time variable is in the argument of the bandlimited impulse response, or the convolution sum where the time variable is in the argument of the function, as an approximation process. In addition to the pointwise and uniform convergence behavior, the convergence behavior in the norm of the considered function space, i.e. the \(L^2\)-norm in our case, is important. While it is well-known that both convolution sums converge uniformly on the whole real axis, the \(L^2\)-norm of the second convolution sum can be divergent for certain functions and systems. We show that the there exist an infinite dimensional closed subspace of functions and an infinite dimensional closed subspace of systems, such that for any pair of function and system from these two sets, we have norm divergence.

In general, the exact distribution of a convolution of independent gamma random variables is quite complicated and does not admit a closed form. Of all the distributions proposed, the gamma...

We determine the optimal linear information and the optimal method of its application to the reconstruction of the convolution of n functions on some convex and centrally symmetric classes of 2 $$ \pi $$ -periodic functions.

In the theory of signal processing, signals are usually classified either by determining whether their time domain is discrete or continuous, or by determining whether they are periodic. However, no comprehensive definitions of multiplication and convolution exist that are consistent with the theories behind all classes, although some important theorems in signal processing involve multiplication and convolution. In order to unite the theories behind these classifications, we will consider tempered distributions. In this paper, we propose an approach to the multiplication and convolution of distributions that is appropriate to signal processing theory, and prove a well-known theorem regarding the impulse response of continuous linear time-invariant systems of tempered distributions in the context of this new approach.

In this article, a convolution of second-order random measures on a locally compact abelian groups is considered. A random measure is an analog of a stochastic process. With a newly defined (random) convolution, we construct a ring and a normed algebra of second-order random measures.