Volume 9, Number 2, August 2008, pp. 215-238
Toshihiko Nishishiraho
Key words:
Approximation process, approximate kernel, absolute moment, convolution type operator, modulus of continuity, stronggly continuous group of operators, summation process, probability density function, summability kernel, Fourier series expansion, multiplier operator, homogeneous Banach space
Mathematices Subject Classification: Primary 41A35, 41A25, 41A65, 44A35

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	Abstract:
	We consider the convergence of approximation processes of convolution type operators in Banach spaces and give quantitative estimates of the rate of their convergence in terms of the modulus of continuity and higher order abosolute moments of approximate kernels. Furthermore, applications are discussed for various summation processes and multiplier operators in connection with Fourier series expansions corresponding to a total, fundamental sequence of mutually orthogonal projections as well as for homogeneous Banach spaces which include the certain classical function spaces, as particular cases. We also give several concrete examples of approximating operators from a probabilistic point of view. These can be induced by various probability density functuions, together with various positive summability kernels.

Appromimation processes of convolutuion type operators in Banach spaces

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