Results 221 to 230 of about 2,918 (250)

Ramanujan’s convolution sum twisted by Dirichlet characters

International Journal of Number Theory, 2019
We find formulas for convolutions of sum of divisor functions twisted by the Dirichlet character [Formula: see text], which are analogous to Ramanujan’s formula for convolution of usual sum of divisor functions. We use the theory of modular forms to prove our results.
Aygin, Zafer Selcuk, Hong, Nankun
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Non-Existence of Convolution Sum System Representations

IEEE Transactions on Signal Processing, 2019
Convolution sum system representations are commonly used in signal processing. It is known that the convolution sum, treated as the limit of its partial sums, can be divergent for certain continuous signals and stable linear time-invariant (LTI) systems, even when the convergence of the partial sums is treated in a distributional setting. In this paper,
Holger Boche, Ullrich J. Mönich, Bernd Meinerzhagen   +2 more
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Distributional Behavior of Convolution Sum System Representations

IEEE Transactions on Signal Processing, 2018
In this paper, we study the validity of the usual convolution sum sampling representation of linear time-invariant (LTI) systems. We consider continuous input signals with finite energy that are absolutely integrable and vanish at infinity. Even for these benign signals, the convolution sum does not always converge.
Holger Boche, Ullrich J. Mönich
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Evaluation of some convolution sums

AIP Conference Proceedings, 2015
We evaluate the convolution sums∑l+50m=nσ(l)σ(m), ∑2l+25m=nσ(l)σ(m), ∑l+25m=nσ(l)σ(m),∑l+m=n,l≡a mod⁡5σ(l)σ(m), for a=0,1,2,3,4using the theory of quasimodular forms.
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CONVOLUTIONS OF RAMANUJAN SUMS AND INTEGRAL CIRCULANT GRAPHS

International Journal of Number Theory, 2012
There exist several generalizations of the classical Dirichlet convolution, for instance the so-called A-convolutions analyzed by Narkiewicz. We shall connect the concept of A-convolutions satisfying a weak form of regularity and Ramanujan sums with the spectrum of integral circulant graphs.
Le, T. A., Sander, J. W.
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Evaluation of certain convolution sums involving the sum of the divisors function

The Ramanujan Journal, 2022
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Evaluate the Convolution Integral and Convolution Sum Use Compact Formula

2011 7th International Conference on Wireless Communications, Networking and Mobile Computing, 2011
Abstract: Convolution integral and convolution summation play an important role in the analysis of the linear time invariant systems. At present, many text books have published in my home country or foreign country , especially the ?sSignals and Systems?t all discuss the methods by use of the graph to determine the up limit, low limit and the interval ...
Bin Ren, De-yong Yu
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Higher convolutions of Ramanujan sums

Journal of Number Theory
Letting \(c_q(n)\) to be the Ramanujan sum, in the paper under review, the authors provide higher convolutions of Ramanujan sums by computing the following limit \[ \lim_{x\to x}\frac{1}{x}\sum_{n\leq x}c_{q_1}(n+a_1)\cdots c_{q_k}(n+a_k). \] The result of the above limit is a multivariable multiplicative function, say \(f(q_1,\dots, q_k)\), for which ...
Goel, Shivani, Murty, M. Ram
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On convoluted sums

Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio computatorica, 2016
Jean-Marie De Koninck, Imre Kátai
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Bounds on shifted convolution sums for Hecke eigenforms

Research in Number Theory, 2022
Morten Risager, Asbjørn Christian Nordentoft, Yiannis N Petridis   +2 more
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