Strong Cesàro summability of double Fourier integrals

Authors: Brown, G.¹; Feng, D.²; Móricz, F.³

Source: Acta Mathematica Hungarica, Volume 115, Numbers 1-2, April 2007 , pp. 1-12(12)

Publisher: Springer

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Abstract:

We prove the following theorem. Assume f ∈ L ^∞(R ²) with bounded support. If f is continuous at some point (x ₁,x ₂) ∈ R ², then the double Fourier integral of f is strongly q-Cesàro summable at (x ₁,x ₂) to the function value f(x ₁,x ₂) for every 0 < q < ∞. Furthermore, if f is continuous on some open subset <EquationSource Format="TEX"><![CDATA[ $$mathcal{G}$$ ]]></EquationSource> of R ², then the strong q-Cesàro summability of the double Fourier integral of f is locally uniform on <EquationSource Format="TEX"><![CDATA[ $$mathcal{G}$$ ]]></EquationSource> .

Keywords: double Fourier transform and integral; inversion formula; partial (or Dirichlet) integral; summability (C, 1); strong q-Cesàro summability; 42B10

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10474-007-4185-z

Affiliations: 1: Email: Vice-Chacellor@vcc.usyd.edu.au 2: Email: dfeng@math.ualberta.ca 3: Email: moricz@math.u-szeged.hu

Publication date: 2007-04-01