TY - JOUR

T1 - Sharp Jackson-type inequalities for approximations of classes of convolutions by entire functions of exponential type

AU - Vinogradov, O. L.

N1 - Funding Information: Acknowledgments. We thank to the Consorcio Minero Benito Juárez, Peña Colorada for supplying the mine products. We also appreciate the kind help of Ernesto Aguilera Torres (Laborato-rio Experimental México de la CFM), Margarita Reyes and Carlos Linares (Laboratorio de Petrología del Instituto de Geofísica, UNAM) and Luis Rendón (Laboratorio Central de Microscopía del Instituto de Física). Finally, we thank the financial support of DGAPA-UNAM research Project IN-108605.

PY - 2006

Y1 - 2006

N2 - In this paper, a new method is introduced for the proof of sharp Jacksontype inequalities for approximation of convolution classes of functions defined on the real line. These classes are approximated by linear operators with values in sets of entire functions of exponential type. In particular, a sharp Jackson-type inequality for the even-order derivatives of the conjugate function is proved. For the uniform and the integral norm, the estimates are sharp even if their left-hand sides are replaced by the best approximation. Sharp inequalities for approximations of periodic functions by trigonometric polynomials and of almost-periodic functions by generalized trigonometric polynomials are special cases of the inequalities mentioned above.

AB - In this paper, a new method is introduced for the proof of sharp Jacksontype inequalities for approximation of convolution classes of functions defined on the real line. These classes are approximated by linear operators with values in sets of entire functions of exponential type. In particular, a sharp Jackson-type inequality for the even-order derivatives of the conjugate function is proved. For the uniform and the integral norm, the estimates are sharp even if their left-hand sides are replaced by the best approximation. Sharp inequalities for approximations of periodic functions by trigonometric polynomials and of almost-periodic functions by generalized trigonometric polynomials are special cases of the inequalities mentioned above.

KW - Entire functions of exponential type

KW - Jackson inequalities

KW - Sharp constants

UR - http://www.scopus.com/inward/record.url?scp=85009812502&partnerID=8YFLogxK

U2 - 10.1090/S1061-0022-06-00922-8

DO - 10.1090/S1061-0022-06-00922-8

M3 - Article

AN - SCOPUS:85009812502

VL - 17

SP - 593

EP - 633

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 4

ER -