Exponential sum
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On the structure of quadratic Gauss sums in the Talbot effect Open
We report on the detailed derivation of the Gauss sums leading to the weighting phase factors in the fractional Talbot effect. In contrast to previous approaches, the derivation is directly based on the two coprime integers p and q that de…
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The hybrid power mean of quartic Gauss sums and Kloosterman sums Open
The main purpose of this paper is using the analytic method and the properties of the classical Gauss sums to study the computational problem of one kind fourth hybrid power mean of the quartic Gauss sums and Kloosterman sums, and give an …
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Computation of best $$L^{\infty }$$ L ∞ exponential sums for 1 / x by Remez’ algorithm Open
The approximation of the function 1 / x by exponential sums has several interesting applications. It is well known that best approximations with respect to the maximum norm exist. Moreover, the error estimates exhibit exponential decay as …
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On the High-Power Mean of the Generalized Gauss Sums and Kloosterman Sums Open
The main aim of this paper is to use the properties of the trigonometric sums and character sums, and the number of the solutions of several symmetry congruence equations to research the computational problem of a certain sixth power mean …
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EXPONENTIAL SUMS OVER PRIMES IN SHORT INTERVALS AND AN APPLICATION TO THE WARING–GOLDBACH PROBLEM Open
Let $\\Lambda(n)$ be the von Mangoldt function, $x$ real and $2\\leq y \\leq x$.\nThis paper improves the estimate on the exponential sum over primes in short\nintervals \\[\n S_k(x,y;\\alpha) = \\sum_{x< n \\leq x+y} \\Lambda(n) e\\left( …
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Kloosterman paths and the shape of exponential sums Open
ISSN:0010-437X
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A new fourth power mean of two-term exponential sums Open
The main purpose of this paper is to use analytic methods and properties of quartic Gauss sums to study a special fourth power mean of a two-term exponential sums mod p , with p an odd prime, and prove interesting new identities. As an app…
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A certain two-term exponential sum and its fourth power means Open
The main purpose of this article is using the properties of the Legendre’s symbol and the classical Gauss sums to study the calculating problem of the fourth power mean of a certain two-term exponential sums, and give an interesting calcul…
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ON THE DISTRIBUTION OF THE MAXIMUM OF CUBIC EXPONENTIAL SUMS Open
In this paper, we investigate the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as ‘Birch sums’. Our main theorem gives upper and lower bounds (of nearly the same order of magnitude) for the …
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One kind sixth power mean of the three-term exponential sums Open
In this paper, we use the estimate for trigonometric sums and the properties of the congruence equations to study the computational problem of one kind sixth power mean of the three-term exponential sums. As a conclusion, we give an exact …
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Application of the AAK theory for sparse approximation of exponential sums Open
In this paper, we derive a new method for optimal $\ell^{1}$- and $\ell^2$-approximation of discrete signals on ${\mathbb N}_{0}$ whose entries can be represented as an exponential sum of finite length. Our approach employs Prony's method …
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The hybrid power mean of the quartic Gauss sums and the two-term exponential sums Open
In this paper, we use the analytic method and the properties of classical Gauss sums to study the computational problems of one kind hybrid power mean of quartic Gauss sums and two-term exponential sums, and give an interesting fourth-orde…
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A Hasse-type principle for exponential Diophantine equations over number fields and its applications Open
In this paper we extend a conjecture (which is a variant of a classical conjecture of Skolem) to exponential Diophantine equations over algebraic number fields. On the one hand, using a generalization of a result of Erdős, Pomerance and Sc…
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On the eighth power mean of the two-term exponential sums Open
The main purpose of this article is using the elementary methods and the quadratic residual transformation technique, and the properties of the solutions of the congruence equations modulo an odd prime p to study the calculating problem of…
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Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums Open
We prove non-trivial bounds for bilinear forms with hyper-Kloosterman sums with characters modulo a prime $q$ which, for both variables of length $M$, are non-trivial as soon as $M\geq q^{3/8+δ}$ for any $δ>0$. This range, which matches Bu…
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On the Hybrid Power Mean of Two-Term Exponential Sums and Cubic Gauss Sums Open
In this paper, an interesting third-order linear recurrence formula is presented by using elementary and analytic methods. This formula is concerned with the calculating problem of the hybrid power mean of a certain two-term exponential su…
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Some Identities Involving Certain Hardy Sums and General Kloosterman Sums Open
Using the properties of Gauss sums, the orthogonality relation of character sum and the mean value of Dirichlet L-function, we obtain some exact computational formulas for the hybrid mean value involving general Kloosterman sums K ( r , l …
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Exact reconstruction of extended exponential sums using rational approximation of their Fourier coefficients Open
In this paper, we derive a new recovery procedure for the reconstruction of extended exponential sums of the form [Formula: see text], where the frequency parameters [Formula: see text] are pairwise distinct. In order to reconstruct y(t) w…
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Difference Sets and Positive Exponential Sums. II: Cubic Residues in Cyclic Groups Open
By constructing suitable nonnegative exponential sums we give upper bounds on the cardinality of any set $B_q$ in cyclic groups $\ZZ_q$ such that the difference set $B_q-B_q$ avoids cubic residues modulo $q$.
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Sums of inverses in thin sets of finite fields Open
We obtain lower bounds for the cardinality of $k$-fold sum-sets of reciprocals of elements of suitable defined short intervals in high degree extensions of finite fields. Combining our results with bounds for multilinear character sums we …
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Mathematical Investigation of Functions Open
Generally, when the independent variable of a given exponential function is used as an exponent, the function is considered an exponential. Thus, the following can be examples of exponential functions: $f(x) = ab^x + c$, $f(x) = ae^bx + c$…
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Exponential bases for partitions of intervals Open
For a partition of $[0,1]$ into intervals $I_1,\ldots,I_n$ we prove the existence of a partition of $\mathbb{Z}$ into $Λ_1,\ldots, Λ_n$ such that the complex exponential functions with frequencies in $ Λ_k$ form a Riesz basis for $L^2(I_k)…
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Squarefree integers in large arithmetic progressions Open
We show that the exponent of distribution of the sequence of squarefree numbers in arithmetic progressions of prime modulus is $\geq 2/3 + 1/57$, improving a result of Prachar from 1958. Our main tool is an upper bound for certain bilinear…
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On discretizing integral norms of exponential sums Open
In this paper we study Lp Marcinkiewicz-Zygmund type inequalitiesc1∑1≤j≤Nwj|g(xj)|p≤‖g‖Lp(K)p≤c2∑1≤j≤Nwj|g(xj)|p for general exponential sums of the form g(x)=∑1≤j≤naje〈λj,x〉,x,λj∈Rd,aj∈R. One of the main results of the paper asserts that …
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Application of the AAK theory for sparse approximation of exponential\n sums Open
In this paper, we derive a new method for optimal $\\ell^{1}$- and\n$\\ell^2$-approximation of discrete signals on ${\\mathbb N}_{0}$ whose entries\ncan be represented as an exponential sum of finite length. Our approach employs\nProny's m…
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The fourth power mean value of one kind two-term exponential sums Open
In this paper, based on the analytic method and the properties of Gauss sums, we study the computational problems of the fourth power mean value of one kind two-term exponential sums through the classification and estimation of Dirichlet c…
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On the sixth power mean of one kind two-term exponential sums weighted by Legendre's symbol modulo $ p $ Open
The main purpose of this article is using the elementary methods and the properties of the character sums of the polynomials to study the calculating problem of one kind sixth power mean of the two-term exponential sums weighted by Legendr…
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A fast Gauss transform in one dimension using sum-of-exponentials approximations Open
We present a fast Gauss transform in one dimension using nearly optimal sum-of-exponentials approximations of the Gaussian kernel. For up to about ten-digit accuracy, the approximations are obtained via best rational approximations of the …
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Coefficients for Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials Open
This is a supplementary dataset for the publication: I. M. Tanash and T. Riihonen, "Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials," in IEEE Transactions on Communications, vol. 68, no. 10, pp.…
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Application of AAK theory for sparse approximation Open
Sparse approximation of structured signals is a common problem in signal processing and system theory. In particular, approximation by exponential sums often arises in natural sciences for the analysis of decay processes. In many applicati…