# Article

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Keywords:
integral modulus of continuity; variation of a function
Summary:
Given a modulus of continuity $\omega$ and $q \in[1, \infty[$ then $H_q^\omega$ denotes the space of all functions $f$ with the period $1$ on $\R$ that are locally integrable in power $q$ and whose integral modulus of continuity of power $q$ (see(1)) is majorized by a multiple of $\omega$. The moduli of continuity $\omega$ are characterized for which $H_q^\omega$ contains "many" functions with infinite "essential" variation on an interval of length $1$.
References:
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[3] J. C. Oxtoby: Mass und Kategorie. Springeг-Verlag, 1971. MR 0393404 | Zbl 0217.09202
[4] A. F. Timan: Theory of Approximation of function of Real Variable. Moskva, 1960. (In Russian.)
[5] G. H. Hardy J. E. Littlewood: Some properties of fractional integrals I, II. Math. Z. 27 (1928), 565-606; З4 (1932), 403-439. MR 1544927

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