We study approximation properties of a family of linear integral operators of Mellin type of the form (T_wf)(s) = \int_{R^N_+} K_w(t)f(st)⟨t⟩^(−1) dt, s ∈ R^N_+ , w > 0, where {K_w}w>0 are bounded approximate identities, ⟨t⟩ := \prod_i=1^N t_i, t = (t_1, . . . , t_N ) ∈ R^N_+ , and f is a function of bounded variation on R^N_+. Here we use a new concept of multidimensional variation in the sense of Tonelli, adapted from the classical definition to the present setting of R^N_+. In particular, a result of convergence in variation is obtained and the problem of the rate of approximation is investigated, with particular attention to the case of Fejér-type kernels.
Variation and approximation in multidimensional setting for Mellin integral operators
ANGELONI, Laura;VINTI, Gianluca
2014
Abstract
We study approximation properties of a family of linear integral operators of Mellin type of the form (T_wf)(s) = \int_{R^N_+} K_w(t)f(st)⟨t⟩^(−1) dt, s ∈ R^N_+ , w > 0, where {K_w}w>0 are bounded approximate identities, ⟨t⟩ := \prod_i=1^N t_i, t = (t_1, . . . , t_N ) ∈ R^N_+ , and f is a function of bounded variation on R^N_+. Here we use a new concept of multidimensional variation in the sense of Tonelli, adapted from the classical definition to the present setting of R^N_+. In particular, a result of convergence in variation is obtained and the problem of the rate of approximation is investigated, with particular attention to the case of Fejér-type kernels.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
