Progress in the Soviet Union on the theory and applications of bandlimited functions

Khurgin, Yakov I.; Yakovlev, Vitaly P.

doi:10.1109/proc.1977.10611

Cited by 26 publications

(29 citation statements)

References 17 publications

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“…, by virtue of (5) and using the identities I ν (z) = i −ν J ν (i z), I ν (−z) = (−1) ν I ν (z) we arrive at (14).…”

Section: I-bessel Sampling Expansions and Sturm-liouville Differentiamentioning

confidence: 99%

“…The historical background of sampling theorems, various applications in many branches of science and engineering, especially in signal analysis and reconstruction and/or its up-to-date results in different areas of mathematics like approximation theory and interpolation are well-covered among others by Jerri's "IEEE 1977 paper" [13], by survey articles of Khurgin-Yakovlev [14] and Unser [24], by the monographs of Higgins [9], an edited monograph by Higgins and Stens [10], the book by Seip [22] and numerous references therein. Thus, by skipping an outline of the facts from the aforementioned references we can focus on our main goal -establishing the I-Bessel sampling expansion result via the appropriate Strum-Liouville boundary value problem and the related sampling expansion series truncation upper bound, which yields the precise convergence rate in this kind of approximation procedures.…”

Section: Introductionmentioning

confidence: 99%

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Sturm–Liouville Problem and I–Bessel Sampling

Maširević¹

2015

APH

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Abstract:The main aim of this article is to establish summation formulae in form of the sampling expansion series building the kernel function by the samples of the modified Bessel function of the first kind I ν , and to obtain a sharp truncation error upper bound occurring in the derived sampling series approximation. Summation formulae for functions I ν+1 /I ν , 1/I ν , I 2 ν and the generalized hypergeometric function 2 F 3 are derived as a by-product of these results. The main derivation tools are the Sturm-Liouville boundary value problem and various properties of Bessel and modified Bessel functions.Keywords: Bessel function of the first kind J ν , modified Bessel function of the first kind I ν , sampling series expansions, Sturm-Liouville boundary value problems, generalized hypergeometric function 2 F 3 , Fox-Wright generalized hypergeometric function p Ψ * q , sampling series truncation error upper bound.

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“…, by virtue of (5) and using the identities I ν (z) = i −ν J ν (i z), I ν (−z) = (−1) ν I ν (z) we arrive at (14).…”

Section: I-bessel Sampling Expansions and Sturm-liouville Differentiamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Sturm–Liouville Problem and I–Bessel Sampling

Maširević¹

2015

APH

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“…(4) can be stated as follows [7]. For each L > 0, there exists an infinite sequence of real eigenfunctions…”

Section: Systems Of Functions With Double Orthogonality and The Genermentioning

confidence: 99%

“…There exist various statements of the uncertainty principle. One of the interpretations is based on the use of eigenfunctions of the truncated Fourier transform, which are functions with double orthogonality [2][3][4][5][6][7]. In the one-dimensional case, these are the so-called prolate spheroidal wave functions, which have been quite comprehensively studied and tabulated [8,9].…”

Section: Introductionmentioning

confidence: 99%

Numerical methods for solving integral equations for functions with double orthogonality

Кравченко

Басараб

2004

Diff Equat

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“…The fact that this is possible was noted by G. Toraldo di Francia in 1952 [12], in applying concepts from super-directive antennas in the microwave community to optical instruments. Closely linked to this phenomenon is the mathematical concept that a function can locally oscillate much faster than its fastest Fourier component [13,14]. The relevance of this mathematical concept to several physical systems was noted by Berry [15], inspired by the work of Aharonov et al on weak measurements in quantum systems [16].…”

Section: Introductionmentioning

confidence: 99%

Optimising superoscillatory spots for far-field super-resolution imaging

et al. 2018

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Optical superoscillatory imaging, allowing unlabelled far-field super-resolution, has in recent years become reality. Instruments have been built and their super-resolution imaging capabilities demonstrated. The question is no longer whether this can be done, but how well: what resolution is practically achievable? Numerous works have optimised various particular features of superoscillatory spots, but in order to probe the limits of superoscillatory imaging we need to simultaneously optimise all the important spot features: those that define the resolution of the system. We simultaneously optimise spot size and its intensity relative to the sidebands for various fields of view, giving a set of best compromises for use in different imaging scenarios. Our technique uses the circular prolate spheroidal wave functions as a basis set on the field of view, and the optimal combination of these, representing the optimal spot, is found using a multi-objective genetic algorithm. We then introduce a less computationally demanding approach suitable for real-time use in the laboratory which, crucially, allows independent control of spot size and field of view. Imaging simulations demonstrate the resolution achievable with these spots. We show a three-order-of-magnitude improvement in the efficiency of focusing to achieve the same resolution as previously reported results, or a 26 % increase in resolution for the same efficiency of focusing. Rev. E 67, 046611 (2003).

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Progress in the Soviet Union on the theory and applications of bandlimited functions

Cited by 26 publications

References 17 publications

Sturm–Liouville Problem and I–Bessel Sampling

Sturm–Liouville Problem and I–Bessel Sampling

Numerical methods for solving integral equations for functions with double orthogonality

Optimising superoscillatory spots for far-field super-resolution imaging

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