A Sinc–Collocation method for the linear Fredholm integro-differential equations

Mohsen, A.; El-Gamel, Mohamed

doi:10.1007/s00033-006-5124-5

Cited by 45 publications

(23 citation statements)

References 24 publications

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“…This follows from the fact that the supports of the terms in (25) are disjoint, by the definition of the sets E(j, q) and relations (20) and (24). In the sequence f q 2 , .…”

Section: Proof Of Lemma 2 2 After the Extension Fmentioning

confidence: 99%

“…Therefore, for every x ∈ (0, π) there exists j = j(x, i) such that, by (20), (24), (25), and (32), the first sum in (37) can be estimated from below as follows:…”

Section: Proof Of Lemma 2 2 After the Extension Fmentioning

confidence: 99%

“…Here, by (19), (20), (24), and (25), for estimating the first sum from below, we can take only part of the terms whose indices correspond to the modes lying from one side of x. We choose the side on which there are at least as many terms as on the opposite side; by (20) and (33), this part has at least [ By (24), every function φ j,q i vanishes at the nodes with even indices that lie in E(j, q i ).…”

Section: Proof Of Lemma 2 2 After the Extension Fmentioning

confidence: 99%

“…As far as I know, before the papers [28]- [35], approximation by such operators on a segment or on a bounded interval had only been studied for certain classes of analytic functions [6,20] by reducing to the case of the axis with the help of a conformal mapping. In [33], an estimate from above was obtained for the best approximation of continuous functions vanishing at the ends of the segment [0, π] by linear combinations of sincs.…”

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confidence: 99%

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On divergence of sinc-approximations everywhere on $(0,\pi)$

Trynin

2011

St. Petersburg Math. J.

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Abstract. Some properties of sinc-approximations of continuous functions on a segment are studied.For the first time, the sinc-approximations arose in the work of Pleynet. Later, in connection with developments in signals coding theory, E. Borel and E. T. Whittaker introduced the notions of a cardinal function and a truncated cardinal function, the restriction of which to [0, π] looks like this:(1)By the present time, the problem of sinc-approximation has been studied rather deeply for functions analytic on the real line and decaying exponentially at infinity (see, e.g.,The most complete survey of the results obtained in this direction before 1993, together with numerous important applications of sinc-approximations, can be found in [6]. An interesting historical survey of investigations on this topic is contained in [7].Also, a series of publications had their origin in what is called the sampling theorem or sometimes, the Whittaker-Kotel nikov-Shannon discretization theorem [8,9,6,11]; in those studies, various representations of entire functions were obtained in the form of series in sincs with interpolation nodes satisfying some "uniform distribution" conditions (see, e.g., [12]-[14]). Starting with the well-known paper [15] by Kramer, the relationship between the sampling theorems and the Lagrange interpolation with nodes in the SturmLiouville problem spectrum has been studied; see, e.g., [16].The sinc-approximations are widely applied for constructing numerical methods of mathematical physics and approximation theory for functions of one or several variables [17]-[21], in the theory of quadrature formulas [6,22] and wavelet transformations [8]-[11], [23,24]. In the paper [25], an analog of the sampling theorem that employs the Hermite type interpolation was obtained.The authors of [26] succeeded in proving approximability for less smooth functions. However, they were forced to modify the operator (1) somewhat. In [26], it was established that if a uniformly continuous and bounded functions f on R belongs to the Dini-Lipchitz class and satisfies the condition f (x) = O(|x| −δ ) as x → ±∞ for some 2010 Mathematics Subject Classification. Primary 42A10, 41A30.

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“…This follows from the fact that the supports of the terms in (25) are disjoint, by the definition of the sets E(j, q) and relations (20) and (24). In the sequence f q 2 , .…”

Section: Proof Of Lemma 2 2 After the Extension Fmentioning

confidence: 99%

“…Therefore, for every x ∈ (0, π) there exists j = j(x, i) such that, by (20), (24), (25), and (32), the first sum in (37) can be estimated from below as follows:…”

Section: Proof Of Lemma 2 2 After the Extension Fmentioning

confidence: 99%

Section: Proof Of Lemma 2 2 After the Extension Fmentioning

confidence: 99%

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On divergence of sinc-approximations everywhere on $(0,\pi)$

Trynin

2011

St. Petersburg Math. J.

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“…where D, G, E, I (0) , I (1) and I (2) are square matrices of order n × n. Particularly, I (0) , I (1) and I (2) are the identity matrix, the skew-symmetric matrix and the symmetric matrix, respectively. For the calculation of unknown coefficients c k in linear system (14), this system should be rewritten by using the notations given in matrix-vector form as…”

Section: Theorem 4 If the Solution Of Problem (1) Is (8) The Determmentioning

confidence: 99%