Direct and Inverse Approximation Theorems for Local Trigonometric Bases

Bittner, Kai; Gröchenig, Karlheinz

doi:10.1006/jath.2002.3688

Cited by 4 publications

(4 citation statements)

References 13 publications

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“…Local approximation operators (Bittner, 2002;Lazzaro and Montefusco, 2002;Renka, 1988a;Zuppa, 2004) are used in data processing (Renka, 1988b(Renka, , 1988c. Two variable Shepard local approximation operators, operators with application in image processing and similar structure data, were introduced (Shepard, 1968).…”

Section: Methodsmentioning

confidence: 99%

IoT Devices Signals Processing Based on Shepard Local Approximation Operators Defined in Riesz MV-Algebras

Noje¹,

Dzițac²,

Pop³

et al. 2020

Informatica

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The Industry 4.0 and smart city solutions are impossible to be implemented without using IoT devices. There can be several problems in acquiring data from these IoT devices, problems that can lead to missing values. Without a complete set of data, the automation of processes is not possible or is not satisfying enough. The aim of this paper is to introduce a new algorithm that can be used to fill in the missing values of signals sent by IoT devices. In order to do that, we introduce Shepard local approximation operators in Riesz MV-algebras for one variable function and we structure the set of possible values of the IoT devices signals as Riesz MV-algebra. Based on these local approximation operators we define a new algorithm and we test it to prove that it can be used to fill in the missing values of signals sent by IoT devices.

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Section: Methodsmentioning

confidence: 99%

IoT Devices Signals Processing Based on Shepard Local Approximation Operators Defined in Riesz MV-Algebras

Noje¹,

Dzițac²,

Pop³

et al. 2020

Informatica

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show abstract

“…is numerically sparse on each interval [22] and only those coefficients with magnitudes exceeding a prescribed threshold need to be stored. Exploiting temporal features present in ray data encountered in the analysis of large scale scattering problems, two types of partitioning schemes are considered.…”

Section: A Representation Of the Ray Data Using Lcbsmentioning

confidence: 99%

“…. The computational cost of the translation stage is dominated by the wavelet-domain translation operation in (22). Although the translation matrix T is highly sparse, its first few columns (corresponding to source LCB functions O nontrivial outgoing rays at each direction in one source box, the overall computational cost of the translation stage scales as (…”

Section: ) Near-normal Incidencementioning

confidence: 99%

A Wavelet-Enhanced PWTD-Accelerated Time-Domain Integral Equation Solver for Analysis of Transient Scattering From Electrically Large Conducting Objects

Liu¹,

Yücel²,

Bağcı³

et al. 2018

IEEE Trans. Antennas Propagat.

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A wavelet-enhanced plane-wave time-domain (PWTD) algorithm for efficiently and accurately solving time-domain surface integral equations (TD-SIEs) on electrically large conducting objects is presented. The proposed scheme reduces the memory requirement and computational cost of the PWTD algorithm by representing the PWTD ray data using local cosine wavelet bases (LCBs) and performing PWTD operations in the wavelet domain. The memory requirement and computational cost of the LCB-enhanced PWTD-accelerated TD-SIE solver, when applied to the analysis of transient scattering from smooth quasi-planar objects with near-normal incident pulses, scale nearly as (log) s s O N N and 1.5 () s O N , respectively. Here, s N denotes the number of spatial unknowns. The efficiency and accuracy of the proposed scheme are demonstrated through its applications to the analysis of transient scattering from a 185 wavelength-long NASA almond and a 123-wavelength long Airbus-A320 model. Index Terms-Fast algorithms, marching-on-in-time (MOT), wavelet, local cosine basis (LCB), plane-wave time-domain algorithm (PWTD), time-domain surface integral equation (TD-SIE), transient scattering, very large-scale problems, complexity analysis.

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“…[9,11,12,14]). For some applications (e.g., in image compression) it is important that a smooth function has a good local approximation by using a very small number of terms, i.e., using only small indices k in (2.6).…”

Section: Reproduction Of Modulated Polynomialsmentioning

confidence: 99%

Formulation of Localized Cosine Bases that Preserve Polynomial Modulated Sinusoids

Bittner

Chui

2004

J. Fourier Anal. Appl.

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We develop a class of compactly supported window functions V m so that each polynomial p(x) of degree m − 1 as well as p(x) modulated with an integer frequency has a locally finite representation in terms of the local cosine basis = {ψ kExplicit formulations for V m and its Fourier transform are derived. It is shown that these window functions are non-negative, have minimal support of length m, and maximal smoothness of order 2m − 2. Furthermore, we determine the exact Riesz bounds of . Smoothness and stability for these bases are superior as compared to other local cosine bases with similar properties in the literature. Consequently, the bases are particularly useful for applications in signal and image processing.

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Direct and Inverse Approximation Theorems for Local Trigonometric Bases

Cited by 4 publications

References 13 publications

IoT Devices Signals Processing Based on Shepard Local Approximation Operators Defined in Riesz MV-Algebras

IoT Devices Signals Processing Based on Shepard Local Approximation Operators Defined in Riesz MV-Algebras

A Wavelet-Enhanced PWTD-Accelerated Time-Domain Integral Equation Solver for Analysis of Transient Scattering From Electrically Large Conducting Objects

Formulation of Localized Cosine Bases that Preserve Polynomial Modulated Sinusoids

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