Topic
Parseval's identity
About: Parseval's identity is a research topic. Over the lifetime, 177 publications have been published within this topic receiving 1758 citations.
Papers published on a yearly basis
1 / 2
Papers
1 Apr 1995
TL;DR: In this article, an intriguing log-cosine integral is fully analyzed and shown to have value a rational multiple of rC(4), C being the Riemann zeta function, and deduce by means of generating functions and Parseval's identity the sums of certain series previously established by a completely different method.
Abstract: An intriguing log-cosine integral is fully analyzed and shown to have value a rational multiple of rC(4), C being the Riemann zeta function. From this we deduce by means of generating functions and Parseval's identity the sums of certain series previously established by a completely different method.
109 citations
TL;DR: In this article, it was shown that the problem of finding a Parseval dual is equivalent to the problem whether a given frame can be dilated to an orthonormal basis (under an oblique projection).
Abstract: Let {x n } be a frame for a Hilbert space H. We investigate the conditions under which there exists a dual frame for {x n } which is also a Parseval (or tight) frame. We show that the existence of a Parseval dual is equivalent to the problem whether (x n ) can be dilated to an orthonormal basis (under an oblique projection). A necessary and sufficient condition for the existence of Parseval duals is obtained in terms of the frame excess. For a frame {π(g)ξ: g ∈ G} induced by a projective unitary representation π of a group G, it is possible that {π(g)ξ: g ∈ G} can have a Parseval dual, but does not have a Parseval dual of the same type. The primary aim of this paper is to present a complete characterization for all the projective unitary representations π such that every frame {π(g)ξ:g ∈ G} (with a necessary lower frame bound condition) has a Parseval dual of the same type. As an application of this characterization together with a result about lattice tiling, we prove that every Gabor frame G(g, L, K) (again with the same necessary lower frame bound condition) has a Parseval dual of the same type if and only if the volume of the fundamental domain of L × K is less than or equal to 1/2.
72 citations
Book•
1 Jan 1982
TL;DR: Fourier Series, Approximation, Singular Integral Operators, and Related TopicsTopics in Analysis and its Applications American journal of mathematicsModern Fourier AnalysisTransactions of the American Mathematical SocietyThe Abel PrizeThe Carleson-Hunt Theorem on Fourier SeriesClassical and Modern Fourier analysis.
Abstract: Fourier SeriesSystems, Approximation, Singular Integral Operators, and Related TopicsTopics in Analysis and Its ApplicationsAmerican journal of mathematicsModern Fourier AnalysisTransactions of the American Mathematical SocietyThe Abel PrizeThe Carleson-Hunt Theorem on Fourier SeriesClassical and Modern Fourier AnalysisAn Introduction to Non-Harmonic Fourier Series, Revised Edition, 93A Course in Functional AnalysisModern Fourier AnalysisHarmonic AnalysisCommutative Harmonic Analysis ICarleson Curves, Muckenhoupt Weights, and Toeplitz OperatorsPointwise Convergence of Fourier SeriesBrownian MotionDifferentiation of Integrals in RnDirichlet SeriesExplorations in Harmonic AnalysisChaos in Classical and Quantum MechanicsMartingales in Banach SpacesFourier AnalysisTrigonometric SeriesClassical Fourier AnalysisThe Geometry of Fractal SetsPerspectives in AnalysisCommutative Harmonic Analysis IVFourier Analysis on Local Fields. (MN-15)Fourier Restriction, Decoupling and ApplicationsWave Packet AnalysisMeasure and IntegralRevue Roumaine de Mathématiques Pures Et AppliquéesFractals in Probability and AnalysisA Panorama of Harmonic AnalysisInterpolation of OperatorsA Course in Abstract Harmonic AnalysisThe Carleson-Hunt theorem on Fourier seriesOrlicz Spaces and Generalized Orlicz SpacesDiophantine Approximation and Dirichlet Series
71 citations
Dissertation•
1 Jan 2013
TL;DR: In this article, the authors present a list of symbology and list of the most commonly used symbols in the English language: list of symbols, list of synonyms, and a reference list of paraphrases.
Abstract: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii OZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
69 citations
TL;DR: In this paper, the Parseval type identities and inequalities for frames in Hilbert spaces were given, which generalize the remarkable results obtained recently by R. Balan, P.G. Kutyniok, D. Casazza, and G. Edidin.
63 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 5 |
| 2020 | 3 |
| 2019 | 2 |
| 2018 | 3 |
| 2017 | 5 |
| 2016 | 6 |