Some Inequalities for the Coefficients in Generalized Fourier Expansions
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Some Inequalities for the Coefficients in Generalized Fourier Expansions Bogdan Gavrea Abstract. In this paper, we derive inequalities for the coefficients in generalized Fourier expansions of (m, n) convex functions in the sense of Popoviciu. Classical Fourier expansions as well as expansions relative to orthogonal polynomials are considered. The results presented here generalize the ones obtained by Niculescu and Rovent¸a (Positivity 24(1):129–139, 2020). Some of the results obtained in deriving inequalities for these coefficients can be further used in obtaining Favard-type inequalities similar to the ones given in Wulbert (Math Comput Model 37(12–13):1383–1391, 2003). Favard type inequalities can be used in obtaining probabilistic inequalities which may be further used in fields such as statistical machine learning. Mathematics Subject Classification. 26D15, 26D05, 46N30. Keywords. Pm,n -simple functionals, polynomials, Favard inequality.
(m, n)-convexity,
Lagrange
1. Introduction In [10], the authors start from the inequality 1 2π f (t) cos(nt)dt ≥ 0, an := π 0 which holds for any convex function on [0, 2π] and any natural number n ≥ 1. The result above is then generalized to the two dimensional box [0, 2π] × [0, 2π]. More precisely, if f : [0, 2π] × [0, 2π] is a continuous function and 2π 2π 1 f (x, y) cos(mx) cos(ny)dxdy, m, n ∈ N∗ am,n := 2 π 0 0 are the cosine Fourier coefficients of function f , in [10], it was proved that am,n ≥ 0, whenever the function f is convex on [0, 2π] × [0, 2π] in the sense of Popoviciu [11] (or (2, 2)-convex in Popoviciu’s terminology). In this work, we generalize the results obtained in [10]. More precisely, we consider the generalized Fourier expansion of a function f : [a, b] × [c, d] → R relative to 0123456789().: V,-vol
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a complete orthogonal system of functions and give sufficient conditions for the coefficients to be non-negative. Let I = [a, b], S be a linear subspace of C([a, b]) and Π be the set of all polynomials with real coefficients. We further assume that Π ⊆ S. The divided difference of a function f ∈ RI on the distinct points x1 , . . . , xn ∈ I is denoted by [x1 , . . . , xn ; f ] and defined by n f (xk ) [x1 , . . . , xn ; f ] = , l (xk ) k=1
where l(x) = (x−x1 ) · · · (x−xn ). The next definition for Pn -simple functionals is due to Popoviciu [12]. Definition 1.1. Let A : S → R be a linear functional. The functional A is called a Pn -simple functional (n ∈ Z, n ≥ −1) if the following hold: (i) A(en+1 ) = 0, where ei : I → R, ei (x) = xi , i ∈ N. (ii) For any function f ∈ S, there exist distinct points ti , ti := ti (f ) ∈ I, i = 1, . . . , n + 2 such that A(f ) = A(en+1 )[t1 , . . . , tn+2 ; f ].
(1)
Remark 1.1. We note that if A is a Pn -simple functional, from (1) we get A(ei ) = 0,
i = 0, 1, . . . , n.
(2)
Definition 1.2. A linear functional A : S → R has the degree of exactness n if condition (i) of Definition 1.1 and (2) are satisfied. The following result was derived by Lupa¸s [7]. A proof can be found in [4]. It states
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