## Functiones et Approximatio Commentarii Mathematici

- Funct. Approx. Comment. Math.
- Volume 34 (2005), 7-17.

### On the Riemann integrability of the $n$-th local modulus of continuity

#### Abstract

The present note proves Riemann integrability of the $n$-th local modulus of continuity which is used within the definition of averaged moduli of smoothness ($\tau$-moduli). In addition it is shown that an averaged supremum norm ($\delta$-norm) can be calculated using the Riemann integral.

#### Article information

**Source**

Funct. Approx. Comment. Math., Volume 34 (2005), 7-17.

**Dates**

First available in Project Euclid: 29 September 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.facm/1538186583

**Digital Object Identifier**

doi:10.7169/facm/1538186583

**Mathematical Reviews number (MathSciNet)**

MR2269660

**Zentralblatt MATH identifier**

1123.26008

**Subjects**

Primary: 26A42: Integrals of Riemann, Stieltjes and Lebesgue type [See also 28-XX]

Secondary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27} 41A25: Rate of convergence, degree of approximation 41A99: None of the above, but in this section

**Keywords**

averaged modulus of smoothness local modulus of continuity Riemann integrability

#### Citation

Goebbels, Steffen J. On the Riemann integrability of the $n$-th local modulus of continuity. Funct. Approx. Comment. Math. 34 (2005), 7--17. doi:10.7169/facm/1538186583. https://projecteuclid.org/euclid.facm/1538186583