Fourier coefficients of continuous functions with respect to localized Haar system

E. S. Belkina^a, Y. V. Malykhin<sup>b

a</sup> Petrozavodsk State University, 33, Lenina pr., Petrozavodsk, Russian Federation
^b Steklov Mathematical Institute, 8, Gubkina st., Moscow, Russian Federation

Abstract: We construct a nontrivial example of a continuous function $f^*$ on $[0,1]^2$ which is orthogonal to tensor products of Haar functions supported on intervals of the same length. This example clarifies the possible behaviour of Fourier coefficients of continuous functions with respect to a localized Haar system. The function $f^*$ has fractal structure. We give lower bounds on its smoothness.

Keywords: fractals, Haar system, Haar wavelets.

UDC: 517.518

MSC: 26B35, 42C10

Received: 15.04.2017
Revised: 15.06.2017
Accepted: 19.06.2017

Language: English

DOI: 10.15393/j3.art.2017.3871

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