Best approximation of constants by polynomials with integer coefficients

What is best approximation of a non-integer number λ∈R by polynomials qn of degree at most n with integer coefficients on the segment [a,b]⊂(0,1) in the uniform metric? In the paper, this old problem is solved for any rational number λ=pq and a segment on the real line. The same problem is solved wh...

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Veröffentlicht in:Journal of approximation theory S. 106225
Hauptverfasser: Trigub, R., Volchkov, V.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier Inc 01.08.2025
Schlagworte:
Algebraic integers
Asymptotic law of distribution of prime numbers
Best approximation of a function
Extremal properties of polynomials
Functional equation for Chebyshev’s polynomials
Integer transfinite diameter
Polynomial with integer coefficients
secondary
Asymptotic law of distribution of prime numbers
Algebraic integers
Extremal properties of polynomials
Polynomial with integer coefficients
Functional equation for Chebyshev’s polynomials
Best approximation of a function
Integer transfinite diameter
primary
ISSN:0021-9045
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Zusammenfassung:What is best approximation of a non-integer number λ∈R by polynomials qn of degree at most n with integer coefficients on the segment [a,b]⊂(0,1) in the uniform metric? In the paper, this old problem is solved for any rational number λ=pq and a segment on the real line. The same problem is solved when the segment is replaced by a disk in ℂ and a cube in Rm, both non containing integer points. Best approximation of rational numbers by polynomials with natural coefficients is considered as well. At the same time, the question of uniqueness and non-uniqueness of best approximation polynomials has also been studied. In addition, connection between theorems on best approximation to functions by polynomials with integer coefficients and integer transfinite diameter is established.
ISSN:0021-9045
DOI:10.1016/j.jat.2025.106225