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Volume 31, Issue 1
Construction Theory of Function on Local Fields

W. Y. Su

Anal. Theory Appl., 31 (2015), pp. 25-44.

Published online: 2017-01

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  • Abstract

We establish the construction theory of function based upon a local field $K_p$ as underlying space. By virtue of the concept of pseudo-differential operator, we introduce "fractal calculus" (or, $p$-type calculus, or, Gibbs-Butzer calculus). Then, show the Jackson direct approximation theorems, Bernstein inverse approximation theorems and the equivalent approximation theorems for compact group $D(\subset K_p)$ and locally compact group $K^+_p(=K_p)$, so that the foundation of construction theory of function on local fields is established. Moreover, the Jackson type, Bernstein type, and equivalent approximation theorems on the Hölder-type space $C^\sigma(K_p), $ $\sigma>0$, are proved; then the equivalent approximation theorem on Sobolev-type space $W^r_\sigma(K_p),$ $ \sigma\geq 0,$ $ 1\leq r<+\infty$, is shown.

  • AMS Subject Headings

41A65, 28A20

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COPYRIGHT: © Global Science Press

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@Article{ATA-31-25, author = {}, title = {Construction Theory of Function on Local Fields}, journal = {Analysis in Theory and Applications}, year = {2017}, volume = {31}, number = {1}, pages = {25--44}, abstract = {

We establish the construction theory of function based upon a local field $K_p$ as underlying space. By virtue of the concept of pseudo-differential operator, we introduce "fractal calculus" (or, $p$-type calculus, or, Gibbs-Butzer calculus). Then, show the Jackson direct approximation theorems, Bernstein inverse approximation theorems and the equivalent approximation theorems for compact group $D(\subset K_p)$ and locally compact group $K^+_p(=K_p)$, so that the foundation of construction theory of function on local fields is established. Moreover, the Jackson type, Bernstein type, and equivalent approximation theorems on the Hölder-type space $C^\sigma(K_p), $ $\sigma>0$, are proved; then the equivalent approximation theorem on Sobolev-type space $W^r_\sigma(K_p),$ $ \sigma\geq 0,$ $ 1\leq r<+\infty$, is shown.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2015.v31.n1.3}, url = {http://global-sci.org/intro/article_detail/ata/4620.html} }
TY - JOUR T1 - Construction Theory of Function on Local Fields JO - Analysis in Theory and Applications VL - 1 SP - 25 EP - 44 PY - 2017 DA - 2017/01 SN - 31 DO - http://doi.org/10.4208/ata.2015.v31.n1.3 UR - https://global-sci.org/intro/article_detail/ata/4620.html KW - Construction theory of function, local field, fractal calculus, approximation theorem, Hölder-type space. AB -

We establish the construction theory of function based upon a local field $K_p$ as underlying space. By virtue of the concept of pseudo-differential operator, we introduce "fractal calculus" (or, $p$-type calculus, or, Gibbs-Butzer calculus). Then, show the Jackson direct approximation theorems, Bernstein inverse approximation theorems and the equivalent approximation theorems for compact group $D(\subset K_p)$ and locally compact group $K^+_p(=K_p)$, so that the foundation of construction theory of function on local fields is established. Moreover, the Jackson type, Bernstein type, and equivalent approximation theorems on the Hölder-type space $C^\sigma(K_p), $ $\sigma>0$, are proved; then the equivalent approximation theorem on Sobolev-type space $W^r_\sigma(K_p),$ $ \sigma\geq 0,$ $ 1\leq r<+\infty$, is shown.

W. Y. Su. (1970). Construction Theory of Function on Local Fields. Analysis in Theory and Applications. 31 (1). 25-44. doi:10.4208/ata.2015.v31.n1.3
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