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Fourier series
About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.
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20 Apr 2009TL;DR: In this article, the authors present a panoramic view of representation theorems in real analysis and their application in the Hilbert transform on R A. The authors also present a survey of real analysis representations.
Abstract: Preface 1. Fourier series 2. Abel-Poisson means 3. Harmonic functions in the unit disc 4. Logarithmic convexity 5. Analytic functions in the unit disc 6. Norm inequalities for the conjugate function 7. Blaschke products and their applications 8. Interpolating linear operators 9. The Fourier transform 10. Poisson integrals 11. Harmonic functions in the upper half plane 12. The Plancherel transform 13. Analytic functions in the upper half plane 14. The Hilbert transform on R A. Topics from real analysis B. A panoramic view of the representation theorems Bibliography Index.
88 citations
TL;DR: In this paper, a general analytical method is derived for the vibration analysis of rectangular plates with elastic edge restraints of varying stiffness, and the displacement solution is sought simply as a linear combination of several one-and two-dimensional Fourier cosine series expansions.
88 citations
TL;DR: In this article, an analytical-Ritz method is developed to study the vibratory characteristics of the elastic thin plate placed into a rectangular hole and connected to the rigid bottom slab of a rectangular container filled with fluid having a free surface.
88 citations
TL;DR: It is shown that a Fourier expansion of the exponential multiplier yields an exponential series that can compute high-accuracy values of the complex error function in a rapid algorithm that is efficient and practically convenient in numerical methods related to the spectral line broadening and other applications requiring errorfunction evaluation over extended input arrays.
88 citations
TL;DR: A survey of work on the mixed problem can be found in this article, where the authors present a formal scheme of the Fourier method for the parabolic equation in a normal cylinder.
Abstract: CONTENTSIntroductionChapter 1. A survey of work on the mixed problem § 1. The formal scheme of the Fourier method § 2. A survey of results contained in the text-books § 3. Investigations relating to the wave equation § 4. The generalised solution of the general hyperbolic equation § 5. Further investigations of the hyperbolic equation § 6. The definition and uniqueness of the classical solution § 7. Solvability of the mixed problem for the hyperbolic equation in an arbitrary normal cylinder § 8. The justification of the Fourier method for the parabolic equation in a normal cylinderChapter 2. Uniqueness of the classical solution in an arbitrary normal cylinder § 9. Uniqueness theorem for the weakly classical solution § 10. Existence of a finite energy for almost all tChapter 3. Convergence of the basic bilinear series § 11. Summary of some results from the theory of elliptic equations § 12. Convergence of the basic bilinear series of eigenfunctions § 13. Convergence of the bilinear series of first derivatives § 14. Convergence of the bilinear series of second derivativesChapter 4. Auxiliary results on the order of magnitude of the Fourier coefficients § 15. Two preliminary lemmas § 16. Basic lemmas on the order of magnitude of the Fourier coefficientsChapter 5. Solvability of the mixed problem for the hyperbolic equation in an arbitrary normal cylinder § 17. Proof of Theorem 8 § 18. Analysis of the conditions of Theorem 8Chapter 6. The justification of the Fourier method for the parabolic equation in an arbitrary normal cylinder § 19. Proof of Theorem 9References
88 citations