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Strong Converse Inequality for a Spherical Operator

By Lin Shaobo and Cao Feilong


<p/> <p>In the paper titled as "Jackson-type inequality on the sphere" (2004), Ditzian introduced a spherical nonconvolution operator <inline-formula> <graphic file="1029-242X-2011-434175-i1.gif"/></inline-formula>, which played an important role in the proof of the well-known Jackson inequality for spherical harmonics. In this paper, we give the lower bound of approximation by this operator. Namely, we prove that there are constants <inline-formula> <graphic file="1029-242X-2011-434175-i2.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2011-434175-i3.gif"/></inline-formula> such that <inline-formula> <graphic file="1029-242X-2011-434175-i4.gif"/></inline-formula> for any <inline-formula> <graphic file="1029-242X-2011-434175-i5.gif"/></inline-formula>th Lebesgue integrable or continuous function <inline-formula> <graphic file="1029-242X-2011-434175-i6.gif"/></inline-formula> defined on the sphere, where <inline-formula> <graphic file="1029-242X-2011-434175-i7.gif"/></inline-formula> is the <inline-formula> <graphic file="1029-242X-2011-434175-i8.gif"/></inline-formula>th modulus of smoothness of <inline-formula> <graphic file="1029-242X-2011-434175-i9.gif"/></inline-formula>.</p

Topics: Mathematics, QA1-939, Science, Q, DOAJ:Mathematics, DOAJ:Mathematics and Statistics
Publisher: Springer
Year: 2011
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