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Realization and Smoothness Related to the Laplacian

Authors:

Z Ditzian K Runovskii

Institution:

(1) Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1

Abstract:

For f

L _n(T ^d) and $\frac{d}{{d + 2}} < p \leqq \infty$ , the modulus of smoothness

$w_\Delta \left( {f,t} \right)_p = \mathop {\sup }\limits_{\left| h \right| \leqq _t } \left\| {2d - \sum\limits_{i = 1}^d {\left( {f\left( { \cdot + he_i } \right) + f\left( { \cdot + he_i } \right)} \right)} } \right\|_p$

is shown to be equivalent to

$\begin{gathered} w_\Delta \left( {f,t} \right)_p = \mathop {\sup }\limits_{\left| h \right| \leqq _t } \left\| {2d - \sum\limits_{i = 1}^d {\left( {f\left( { \cdot + he_i } \right) + f\left( { \cdot + he_i } \right)} \right)} } \right\|_p \hfill \\ \left\| {f - T_n } \right\|_p + n^{ - 2} \left\| {\Delta T_n } \right\|_p ,n = \left {\frac{1}{t}} \right] + 1 \hfill \\ \end{gathered}$

where T _n is the best trigonometric polynomial approximant of degree n to f in L _p and Delta

is the Laplacian. The above result is shown to be incorrect for 0 < p

$\frac{d}{{d + 2}}$ .

Keywords:

realization discrete Laplacian Bernstein type inequalities Jackson-type inequalities

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