On the Fourier transforms of self-similar measures

Abstract

For the Fourier transform of a general (non-trivial) self-similar measure μ on the real line , we prove a large deviation estimate

Keywords:

• self-similar measure
• Fourier transform
• large deviation

2010 Mathematics Subject Classification:

• Primary 28A80
• Secondary 42A38

Acknowledgements

This work was supported by JSPS KAKENHI, Grant Number 22340035.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. Actually we cannot apply the results in [10] and [11] directly because the function ψ has logarithmic singularities at x = ±π/2 and hence is not continuous. Still we can obtain the large deviation estimate stated here by approximating ψ by smooth functions from above. It seems reasonable to expect that the limit is exact and the inequality is actually an equality.

Additional information

Funding

This work was supported by JSPS KAKENHI [grant number 22340035].