Introduction to Fourier Analysis on Euclidean Spaces (PMS-32)

Stein, Elias M.; Weiss, Guido

doi:10.1515/9781400883899

Cited by 2,032 publications

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“…First of all we observe that condition (5) says that K * is of restricted weak type p for 1 < p ≤ 2. Therefore, well known results in interpolation theory (see [12]) tell us that it is also of strong type p for 1 < p < 2. That is,…”

Section: A General Approximation Principlementioning

confidence: 92%

Remarks on a theorem by N. Yu. Antonov

Sjölin

Soria

2003

Studia Math.

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Abstract.We extend some results of N. Yu. Antonov on convergence of Fourier series to more general settings. One special feature of our work is that we do not assume smoothness for the kernels in our hypotheses. This has interesting applications to convergence with respect to general orthonormal systems, like the Walsh-Fourier system, for which we prove a.e. convergence in the class L log L log log log L. Other applications are given in the theory of differentiation of integrals.

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Section: A General Approximation Principlementioning

confidence: 92%

Remarks on a theorem by N. Yu. Antonov

Sjölin

Soria

2003

Studia Math.

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“…For details about the Riesz transformation, refer to [3,29]. By the boundedness of the Riesz operator R in the BMO space, we obtain…”

Section: Morrey Spacesmentioning

confidence: 99%

New regularity criteria for the 3D Hall-MHD equations

2018

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“…We get the following. (1) , (0) , and (1) be stated in Section 2. Then, for their Fourier coefficients, one has the following asymptotic formulas:…”

Section: Asymptotic Formula Of the Errors For The Reconstruction Of (mentioning

confidence: 99%

“…It is well-known that smooth periodic functions can be approximated well by Fourier series [1,2]. But, if we expand a bivariate function on the square robustly into Fourier series, in general, its Fourier coefficients decay slowly.…”

Section: Introductionmentioning

confidence: 99%

Approximation of Functions on a Square by Interpolation Polynomials at Vertices and Few Fourier Coefficients

Zhang

2017

Journal of Function Spaces

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For a bivariate function on a square, in general, its Fourier coefficients decay slowly, so one cannot reconstruct it by few Fourier coefficients. In this paper we will develop a new approximation scheme to overcome the weakness of Fourier approximation. In detail, we will use Lagrange interpolation and linear interpolation on the boundary of the square to derive a new approximation scheme such that we can use the values of the target function at vertices of the square and few Fourier coefficients to reconstruct the target function with very small error.

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Introduction to Fourier Analysis on Euclidean Spaces (PMS-32)

Cited by 2,032 publications

References 0 publications

Remarks on a theorem by N. Yu. Antonov

Remarks on a theorem by N. Yu. Antonov

New regularity criteria for the 3D Hall-MHD equations

Approximation of Functions on a Square by Interpolation Polynomials at Vertices and Few Fourier Coefficients

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