A scheme for interpolation by Hankel translates of a basis function

作者：Arteaga Cristian; Marrero Isabel^*

来源：Journal of Approximation Theory, 2012, 164(12): 1540-1576.

DOI：10.1016/j.jat.2012.08.005

• Basis function
• Bessel-Kingman hypergroup
• Besse%26apos
• operator
• Hankel convolution
• Hankel translation
• Minimal norm interpolant
• Sobolev embedding theorem

This work discusses interpolation of complex-valued functions defined on the positive real axis I by certain special subspaces, in a variational setting that follows the approach of Light and Wayne [W. Light, H. Wayne, Spaces of distributions, interpolation by translates of a basis function and error estimates, Numer. Math. 81 (1999) 415-450]. The set of interpolation points will be a subset {a(1), ... , a(n)} of I and the interpolants will take the form %26lt;br%26gt;u(x) = Sigma(n)(i=1)alpha(i)(tau(ai)phi)(x) + Sigma(m-1)(j=0) beta(j)p(mu,j)(x) (x is an element of I), %26lt;br%26gt;where mu %26gt;= -1/2, phi is a complex function defined on I (the so-called basis function), p(mu,j) (x) = x(2j+mu+1/2) (j is an element of Z(+), 0 %26lt;= j %26lt;= m - 1) is a Muntz monomial, tau(z) (z is an element of I) denotes the Hankel translation operator of order mu, and alpha(i), beta(j) (i, j is an element of Z(+), 1 %26lt;= i %26lt;= n, 0 %26lt;= j %26lt;= m - 1) are complex coefficients. An estimate for the pointwise error of these interpolants is given. Some numerical examples are included.

• 出版日期2012-12