223 research outputs found

    Fractional Hankel and Bessel wavelet transforms of almost periodic signals

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    The main objective of this paper is to study the Hankel, fractional Hankel, and Bessel wavelet transforms using the Parseval relation. We construct a generalized frame and write new relations and inequalities using almost periodic functions, strong limit power signals, and these transform methods.Publisher's Versio

    Elastic Wave Radiation from a Line Source of Finite Length

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    Integral Transformation, Operational Calculus and Their Applications

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    The importance and usefulness of subjects and topics involving integral transformations and operational calculus are becoming widely recognized, not only in the mathematical sciences but also in the physical, biological, engineering and statistical sciences. This book contains invited reviews and expository and original research articles dealing with and presenting state-of-the-art accounts of the recent advances in these important and potentially useful subjects

    Sampling Sparse Signals on the Sphere: Algorithms and Applications

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    We propose a sampling scheme that can perfectly reconstruct a collection of spikes on the sphere from samples of their lowpass-filtered observations. Central to our algorithm is a generalization of the annihilating filter method, a tool widely used in array signal processing and finite-rate-of-innovation (FRI) sampling. The proposed algorithm can reconstruct KK spikes from (K+K)2(K+\sqrt{K})^2 spatial samples. This sampling requirement improves over previously known FRI sampling schemes on the sphere by a factor of four for large KK. We showcase the versatility of the proposed algorithm by applying it to three different problems: 1) sampling diffusion processes induced by localized sources on the sphere, 2) shot noise removal, and 3) sound source localization (SSL) by a spherical microphone array. In particular, we show how SSL can be reformulated as a spherical sparse sampling problem.Comment: 14 pages, 8 figures, submitted to IEEE Transactions on Signal Processin

    Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets

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    We prove a Calder\'on reproducing formula for the Dunkl continuous wavelet transform on R\mathbb{R}. We apply this result to derive new inversion formulas for the dual Dunkl-Sonine integral transform

    Fourier Transforms

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    The 21st century ushered in a new era of technology that has been reshaping everyday life, simplifying outdated processes, and even giving rise to entirely new business sectors. Today, contemporary users of products and services expect more and more personalized products and services that can meet their unique needs. In that sense, it is necessary to further develop existing methods, adapt them to new applications, or even discover new methods. This book provides a thorough review of some methods that have an increasing impact on humanity today and that can solve different types of problems even in specific industries. Upgrading with Fourier Transformation gives a different meaning to these methods that support the development of new technologies and have a good projected acceleration in the future

    Predicted Deepwater Bathymetry From Satellite Altimetry: Non-Fourier Transform Alternatives

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    Robert Parker (1972) demonstrated the effectiveness of Fourier Transforms (FT) to compute gravitational potential anomalies caused by uneven, non-uniform layers of material. This important calculation relates the gravitational potential anomaly to sea-floor topography. As outlined by Sandwell and Smith (1997), a six-step procedure, utilizing the FT, then demonstrated how satellite altimetry measurements of marine geoid height are inverted into seafloor topography. However, FTs are not local in space and produce Gibb’s phenomenon around discontinuities. Seafloor features exhibit spatial locality and features such as seamounts and ridges often have sharp inclines. Initial tests compared the windowed-FT to wavelets in reconstruction of the step and saw-tooth functions and resulted in lower Root Mean Square (RMS) error with fewer coefficients. This investigation, thus, examined the feasibility of utilizing sparser base functions such as the Mexican Hat Wavelet, which is local in space, to first calculate the gravitational potential, and then relate it to sea-floor topography, with the aim of improving satellite derived bathymetry maps

    UMD-valued square functions associated with Bessel operators in Hardy and BMO spaces

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    We consider Banach valued Hardy and BMO spaces in the Bessel setting. Square functions associated with Poisson semigroups for Bessel operators are defined by using fractional derivatives. If B is a UMD Banach space we obtain for B-valued Hardy and BMO spaces equivalent norms involving γ\gamma-radonifying operators and square functions. We also establish characterizations of UMD Banach spaces by using Hardy and BMO-boundedness properties of g-functions associated to Bessel-Poisson semigroup
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