Effective dimension for weighted function spaces
نویسنده
چکیده
This paper introduces some notions of effective dimension for weighted function spaces. A space has low effective dimension if the smallest ball in it that contains a function of variance 1, has no functions with large values of certain ANOVA mean squares. For a Sobolev space of periodic functions defined by product weights we get explicit formulas describing effective dimension in terms of those weights. In particular, for a space with product weights it is possible to compute truncation and superposition dimensions directly from the weight sequence. For weights γj = γ1j −q with q > 1, and γj 6 1, the result is a low superposition dimension though high truncation dimensions are possible.
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