On Möbius’ inversion formula and closed sets of functions

Möbius inversion formula for monoids with zero

The Möbius inversion formula, introduced during the 19th century in number theory, was generalized to a wide class of monoids called locally finite such as the free partially commutative, plactic and hypoplactic monoids for instance. In this contribution are developed and used some topological and algebraic notions for monoids with zero, similar to ordinary objects such as the (total) algebra o...

Multiplicative Functions and Möbius Inversion Formula

Examples. Let n ∈ N. Define functions τ, σ, π : N → N as follows: • τ(n) = the number of all natural divisors of n = #{d > 0 | d|n}; • σ(n) = the sum of all natural divisors of n = ∑ d|nd; • π(n) = the product of all natural divisors of n = ∏ d|nd. As we shall see below, τ and σ are multiplicative functions, while π is not. From now on we shall write the prime decomposition of n ∈ N as n = p1 1...

On Pre-θ-Open Sets and Two Classes of Functions

The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials

Hurwitz found the Fourier expansion of the Bernoulli polynomials over a century ago. In general, Fourier analysis can be fruitfully employed to obtain properties of the Bernoulli polynomials and related functions in a simple manner. In addition, applying the technique of Möbius inversion from analytic number theory to Fourier expansions, we derive identities involving Bernoulli polynomials, Ber...

The Calculation of Fourier Coefficients by the Möbius Inversion of the Poisson Summation Formula

In Part I, the MIPS method for calculating Fourier coefficients was introduced, and applied to functions / £ C(p>[0, 1]. In this part two extensions of the theory are described. One modification extends the theory to piecewise continuous functions, / £ PC(p'[0, 1]. Using these results the method may be used to calculate approximations to trigonometrical integrals (in which the length of the int...