Chebyshev polynomials for higher order differential equations and fractional powers

• Published in: Mathematische annalen Vol. 388; no. 1; pp. 675 - 702
• Main Authors: Bezerra, Flank D. M., Santos, Lucas A.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 2024; Springer Nature B.V
• Subjects: Cauchy problems; Chebyshev approximation; Differential equations; Hilbert space; Initial conditions; Mathematics; Mathematics and Statistics; Operators (mathematics); Polynomials; 47A08; 35R11; 35K52
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-022-02554-x

In this paper we show the characterization of the fractional powers of a class of positive operators by Chebyshev polynomials of the second kind. We consider the following higher order abstract Cauchy problems 0.1 d n u d t n + A u = 0 , t > 0 , with initial conditions given by d i u d t i ( 0 ) = u i ∈ X n - ( i + 1 ) n , i ∈ { 0 , 1 , … , n - 1 } , n ⩾ 1 , where X be a separable Hilbert space and A : D ( A ) ⊂ X → X is an unbounded linear, closed, densely defined, self-adjoint and positive definite operator, and its fractional counterpart. Here, X α ( 0 ⩽ α ⩽ 1 ) denotes the domain of the fractional powers A α endowed with graphic norm.