DISCRETE d-DIMENSIONAL MODULI OF SMOOTHNESS

作者：Ditzian Z^*; Prymak A

来源：PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2014, 142(10): 3553-3559.

DOI：10.1090/S0002-9939-2014-12088-0

• Moduli of smoothness
• diadic mesh
• tensor product splines
• Lagrange interpolation

We show that on the d-dimensional cube I-d = [0, 1](d) the discrete moduli of smoothness which use only the values of the function on a diadic mesh are sufficient to determine the moduli of smoothness of that function. As an important special case our result implies for f is an element of C(I-d) and a given integer r that when 0 %26lt; alpha %26lt; r, the condition %26lt;br%26gt;vertical bar Delta(r)(2-n)(ei)f(k(1)/2(n),...,k(d)/2(n))vertical bar %26lt;= M2(-n alpha) %26lt;br%26gt;for integers 1 %26lt;= i %26lt;= d, 0 %26lt;= k(i) %26lt;= 2(n) - r, 0 %26lt;= k(j) %26lt;= 2(n) when j not equal i, and n = 1, 2,... is equivalent to %26lt;br%26gt;vertical bar Delta(r)(he)f(xi)vertical bar %26lt;= M(1)h(alpha) for xi, e is an element of R-d, h %26gt; 0 and vertical bar e vertical bar = 1 such that xi, xi+rhe is an element of I-d.

• 出版日期2014-10