Strong Converse Inequality for a Spherical Operator
<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...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2011-01-01
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| Series: | Journal of Inequalities and Applications |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/2011/434175 |
| Summary: | <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> |
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| ISSN: | 1025-5834 1029-242X |