POSITIVE DEFINITE FUNCTIONS AND SHARP INEQUALITIES FOR PERIODIC FUNCTIONS

Let φ be a positive definite and continuous function on R, and let μ be the corresponding Bochner measure. For fixed ε, τ ∈ R, ε 6= 0, we consider a linear operator Aε,τ generated by the function φ: Aε,τ (f)(t) := ∫ R ef(t + εu)dμ(u), t ∈ R, f ∈ C(T). Let J be a convex and nondecreasing function on [0,+∞). In this paper, we prove the inequalities ‖Aε,τ (f)‖p 6 φ(0)‖f‖p , ∫ T J (|Aε,τ (f)(t)|) dt ≤ ∫


Introduction
The role of positive definite functions in obtaining sharp inequalities for trigonometric polynomials and entire functions is well known (see, for instance, Boas [6,Ch. 11], Timan [22,Sect. 4.8], Lizorkin [13], Gorin [9], and Trigub and Belinsky [23]).For instance, the classical Bernstein inequality max |f ′ (x)| ≤ n max |f (x)| for trigonometric polynomials of degree at most n is related to the positive definiteness of the function (1 − |x|) + .A historical survey of such inequalities and the methods of their proof are given in the works by Gorin [9], Arestov and Glazyrina [5], Gashkov [8], and Vinogradov [25].In the present paper, we obtain sharp inequalities for continuous periodic functions and, in particular, for trigonometric polynomials.These inequalities are related to positive definite functions.As consequences, we obtain generalizations of Bernstein-Szegő inequalities.We give criteria and descriptions of extremal functions in these inequalities.
A complex-valued function f : R → C is called positive definite on R (f ∈ Φ(R)) if, for any m ∈ N, any set of points {x k } m k=1 ⊂ R, and any complex numbers {c k } m k=1 ⊂ C, the following inequality holds: It is easy to verify that, for any β ∈ R, the function f (x) = e iβx is positive definite.For a function in Φ(R), the continuity at zero is equivalent to the continuity on R. If f, g ∈ Φ(R), then |f (x)| ≤ f (0), f (−x) = f (x), |f (x+y)−f (x)| 2 ≤ 2f (0)(f (0)−Ref (y)), x, y ∈ R, and f , Re f , f g ∈ Φ(R).In 1932, S. Bochner and, independently, A. Khinchin proved the following criterion of positive definiteness.
Theorem 1 (Bochner-Khinchin).The inclusion f ∈ Φ(R) ∩ C(R) holds if and only if there exists a finite nonnegative Borel measure µ on R such that The proof of this theorem can be found, for instance, in [2,7,19,23,24].As a direct consequence, we obtain the following criterion of positive definiteness in terms of nonnegativity of the Fourier transform: Using this criterion, it is not difficult to see that the functions (1−|x|) + , e −|x| , and e −x 2 are positive definite.
We study in more detail the case in which ε = 1/n, n ∈ N, τ = 1, and ϕ(x) ≡ e iβx ψ(x), where β ∈ R and ψ is a 2-periodic function of the class Φ(R) ∩ C(R) (see Theorem 5 and Remarks 4 and 5).In turn, we consider in more detail the case where a 2-periodic function ψ is constructed by means of a finite function g ∈ Φ(R) ∩ C(R) (Theorem 6).As a particular case, we obtain the Berstein-Szegő inequality for the Weyl-Nagy derivative of trigonometric polynomials (Remark 7).In Theorem 8, we consider the case of the family of functions g 1/n,h (x) := hg(x) and the function g ∈ C(R) is even, nonnegative, decreasing, and convex on (0, +∞) with supp g ⊂ [−1, 1].This case is related to the positive definiteness of piecewise linear functions [15].In Theorem 9 and Corollary 3, we obtain general interpolation formulas for periodic functions which include the known interpolation formulas of M. Riesz, of G. Szegő, and of A.I. Kozko [11] for trigonometric polynomials (see Remark 8).

Auxiliary facts of measure and integration theory
We recall some well-known facts which are used in the paper to describe extremal functions.In this section, a measure µ is a nonnegative countably additive function defined on a σ-algebra γ with identity element Ω.For p ∈ (0, +∞), the class L p (Ω, γ, µ) is the set of all γ-measurable functions f : Ω → C such that The class L ∞ (Ω, γ, µ) is the set of all γ-measurable functions f : Ω → C for which there exists For convenience, we assume that and the inequality turns into an equality if and only if the equality f (u) = e iθ |f (u)| holds for some θ ∈ R and for µ-almost all u ∈ Ω. P r o o f.See, for instance, [18, Theorems 1.33 and 1.39].Obviously, for some β ∈ R, we have and the inequality turns into an equality if and only if Re(e iβ f (u Proposition 2. Assume that J is a convex function on R, (Ω, γ, µ) is a measurable space with finite measure, µ(Ω) > 0, and f is a real-valued function in L 1 (Ω, µ).Then (2.1) For a proof of this result, see, for instance, [12,Sect. 2.2].
The next proposition will be needed only in Remark 3.
(ii) If f q = 0, the required assertion is obvious.Let f q > 0.Then, for any p > q, the inequality f p ≤ f (p−q)/p ∞ f q/p q holds.This inequality and assertion (i) yield lim sup

Sharp L p -inequalities for periodic functions
Equality (1.1) implies the inequality It follows from the Fubini theorem that the Fourier series of the function A ε,τ (f )(t) has the form where c k (f ) are the Fourier coefficients of the function f : Let us find sufficient conditions for the equality If |ϕ(εs − τ )| = ϕ(0) for some s ∈ Z, then equality (3.3) holds for the polynomial If, for some s, m ∈ Z, s = m, we have then equality (3.3) holds for the polynomial f (t) = ce ist + νe imt , c, ν ∈ C, since, in this case, We only need to take into account that, for any δ, α ∈ R, the following equalities hold: In particular, the latter equality holds for α = δ/(s − m).
Thus, we have proved the following theorem.
When p = 1, inequality (3.5) turns into an equality at some function f ∈ C(T) (see inequality (3.1) and Proposition 1) if and only if, for any t ∈ R, there exists a number β(t) ∈ R such that the equality f (t + εu) = e i(uτ +β(t)) |f (t + εu)| holds for µ-almost all u ∈ R.This implies that if a function f ∈ C(T) is extremal in inequality (3.5) with p = 1, then any function of the form cf (t)g(t), where c ∈ C, g ∈ C(T), and g(t) ≥ 0 for all t ∈ R, is also extremal.
When p ∈ (1, ∞), inequality (3.5) turns into an equality at some function f ∈ C(T) if and only if, for any t ∈ R and µ-almost all u ∈ R, the equality f (t + εu) = e iuτ c(t) holds, where is everywhere dense in L p (T) (the Lebesgue measure is taken as a measure).Therefore, inequality (3.5) implies that the multiplier A ε,τ : C(T) → C(T) defined by formula (3.2) is extended to the multiplier A ε,τ : L p (T) → L p (T), 1 ≤ p < ∞, and and inequality (3.6) holds with p = ∞.We only need to use the well-known facts from measure and integration theory (see Proposition 3).

Periodic positive definite functions
The following description of periodic functions of the class Φ(R) ∩ C(R) is well known (see, for instance, [7, Theorem 1.7.5] and [10, Sect.II.1]).
In this case, the function ψ is expanded into the absolutely convergent Fourier series , and a 2-periodic function ψ(x) coincides with the function f (x) Only terms with k = 0 and k = 1 remain in this sum for x ∈ [−2, 2].

Sharp integral inequalities for periodic functions
Let ϕ ∈ Φ(R) ∩ C(R) and ϕ(0) > 0. Assume that J is a convex nondecreasing function on [0, +∞).Then J is continuous on [0, +∞) and can be extended to R with preservation of convexity (for instance, by defining J(t) := J(0) for t < 0 or by means of the even extension).
Successively using the monotonicity and the Jensen inequality (see, for instance, [12, Sect.2.2] or Proposition 2), for f ∈ C(T), we derive from inequality (3.1) that We integrate the left-hand and right-hand sides of inequality (5.1) with respect to t ∈ T. Applying the Fubini theorem and taking into account the periodicity of f , we obtain In view of the arbitrariness of f , it is convenient to write the latter inequality in the form Inequality (5.2) also holds if ϕ(0) = 0, since, in this case, ϕ(x) ≡ 0 and, hence, A ε,τ (f )(t) ≡ 0 for any f ∈ C(T).Thus, we obtain the following theorem.
If condition (3.4) holds for some s, m ∈ Z, s = m, then equality in (5.2) is attained at the polynomials 3) If the function J is strictly convex at any point of the interval (0, +∞) and ϕ(0) > 0, then inequality (5.2) turns into an equality at some function f ∈ C(T) if and only if, for any t ∈ R and µ-almost all u ∈ R, the equality e −iuτ f (t + εu) = c(t) holds, where c(t) = A ε,τ (f )(t)/ϕ(0) ∈ C(T).P r o o f.Only the latter statement needs to be proved.The sufficiency is obvious.Let us prove the necessity.Let inequality (5.2) turn into an equality for some function f ∈ C(T).Then inequalities (5.1) turn into equalities for all t ∈ R. Let Proposition 2).Since the function J strictly increases on [0, +∞), inequality (3.1) also turns into an equality for all t ∈ R. Therefore, for some β(t) ∈ R and µ-almost all u ∈ R, we have the equality (see Proposition 1) For ε = 1/n, n ∈ N, and τ = 1, we can distinguish the case where the condition on the extremal function in Theorem 4 is more clear.Theorem 5. Let ϕ(x) ≡ e iβx ψ(x), where β ∈ R, and let ψ be a 2-periodic function in Φ(R) ∩ C(R).Let J be a convex nondecreasing function on [0, +∞).Then the operator A 1/n,1 , n ∈ N, generated by the function ϕ by formula (1.1) for ε = 1/n and τ = 1 satisfies the inequality (5.3) 3) turns into an equality, in particular, at every function f ∈ C(T) whose Fourier series has the form If the function J is strictly convex at any point of the interval (0, +∞) and ψ(0) > 0, then inequality (5.3) turns into an equality at some function f ∈ C(T) if and only if the functions (−1) s f t + πs n are identical on R for all s = 0, . . ., 2n − 1 such that µ s (n, ψ) > 0, where and c k (ψ) ≥ 0, k ∈ Z, are the Fourier coefficients of the function ψ.If, in addition, the inequalities µ s (n, ψ) > 0 and µ s+1 (n, ψ) > 0 hold for some s ∈ Z, then inequality (5.3) turns into an equality only at functions f ∈ C(T) whose Fourier series has the form (5.4).
Since the function ψ belongs to Φ(R) ∩ C(R) and is 2-periodic, its Fourier coefficients c k (ψ), k ∈ Z, are nonnegative and ψ is expanded into an absolutely convergent Fourier series.Then the function ϕ is also expanded into an absolutely convergent series: It follows from this representation that the Bochner measure µ of the function ϕ is concentrated at the points Taking into account the periodicity of f , it is convenient to divide the terms in this sum into disjoint groups in which the summation index has the form k + 2nm with m ∈ Z and k = 0, . . ., 2n − 1.Then where the numbers µ k (n, ψ) are defined by formula (5.5).For these numbers, the following equalities hold: If a function f belongs to C(T) and its Fourier series has the form (5.4), then, obviously, (−1) s f (t + πs/n) ≡ f (t) for all s ∈ Z.Therefore, for such functions, we have A 1/n,1 (f )(t) ≡ e −iβ ψ(0)f (t + β/n) and inequality (5.3) turns into an equality.
If the function J is strictly convex at any point of the interval (0, +∞) and ψ(0) > 0, then Theorem 4 implies that inequality (5.3) turns into an equality at some function f ∈ C(T) ⇐⇒ the functions (−1) s f (t + (πs + β)/n) are identical on R for all s ∈ Z such that µ({t s }) = c s (ψ) > 0 ⇐⇒ the functions (−1) s f (t + πs/n) are identical on R for all s = 0, . . ., 2n − 1 such that µ s (n, ψ) > 0. The latter equivalence is a consequence of the following properties: (1) the functions of this family with numbers s ∈ Z and s + 2nm, m ∈ Z, are identical; (2) Assume that inequality (5.3) turns into an equality at some function f ∈ C(T).If, in addition, the inequalities µ s (n, ψ) > 0 and µ s+1 (n, ψ) > 0 hold for some s ∈ Z, then, by what has been proved, Then, for the Fourier coefficients of the function f , we have the equalities for some m ∈ Z.This means that the Fourier series of the function f has the form (5.4).The theorem is proved.
2) When p = 1, inequality (5.8) turns into an equality at some function f ∈ C(T) if and only if, for any t ∈ R, there exists a number δ(t) ∈ R such that the identity (−1) s f t + πs n ≡ e iδ(t) f t + πs n (5.10) holds for all s = 0, . . ., 2n − 1 such that µ s (n, ψ) > 0. This implies that if a function f ∈ C(T) is extremal in inequality (5.8) with p = 1, then any function of the form cf (t)g(t), where c ∈ C, g ∈ C(T), and g(t) ≥ 0 for all t ∈ R, is also extremal.In particular, functions of the form h(t)g(t) are extremal if the function h ∈ C(T) has the form (5.4), g ∈ C(T), and g(t) ≥ 0 for all t ∈ R.
If µ s (n, ψ) > 0 for s = 0, . . ., 2n − 1, then only polynomials of the form f (t) = ce int + νe −int , c, ν ∈ C, are extremal among trigonometric polynomials of degree at most n for which inequality (5.8) with p = 1 turns into an equality.Indeed, if f is an extremal trigonometric polynomial of degree at most n, then condition (5.10) is satisfied for s = 0, . . ., 2n − 1.Then one can use the Riesz interpolation formula [16,17] (see also [28, Ch.X, Sect.3, (3.11)]) , where all a s > 0 and Remark 5. If, in Theorem 5, the function J is convex and strictly increasing on [0, +∞) and µ s (n, ψ) > 0 for all s = 0, . . ., 2n − 1 (this implies that ψ(0) > 0), then only polynomials of the form f (t) = ce int + νe −int , c, ν ∈ C, are extremal among trigonometric polynomials of degree at most n for which inequality (5.3) turns into an equality.Indeed, if inequality (5.3) turns into an equality at some function f ∈ C(T), then the corresponding inequalities (5.1) and (3.1) turn into equalities for any t ∈ R and, hence, inequality (5.8) with p = 1 turns into an equality at f .Then we need to use the last statement in Remark 4.
In conclusion of this section, we note that the integral inequalities (5.2) for the class of trigonometric polynomials and for different differential operators and Szegő compositions were studied by many authors, in particular, by A. Zygmund, V.V. Arestov, V.I.Ivanov, E.A. Storozhenko, V.G.Krotov, P. Oswald, and A.I. Kozko.In this case, not only convex functions J were considered.A history of this question was described in great detail in the paper by Arestov [4].

Generalization of Bernstein-Szegő inequalities
We denote by F n , n ∈ N, the set of trigonometric polynomials For β = rπ/2, we obtain the Weyl derivative which, for r ∈ N, coincides with the usual derivative of order r.Often, this operator is called the Weyl-Nagy derivative.
Other cases in which inequality (6.2) holds, when r < 1 or 0 ≤ p < 1, were considered in the paper by Arestov and Glazyrina [5], where these inequalities are called Bernstein-Szegő inequalities and a complete history of such inequalities is given.
Inequalities more general than (6.1) and (6.2) are obtained from Theorem 5 under an appropriate choice of the function ψ.The method of construction of the function ψ described below is essentially contained in the paper by Lizorkin [13].
The case where r = 1, β = πq with q ∈ Z, n ≥ 2, and p = 1 or p = ∞ has not been studied.
One can use the positive definite function g α,h given in Corollary 2 to obtain new sharp inequalities for trigonometric polynomials.
Theorem 8. Let a function g ∈ C(R) be even, nonnegative, decreasing, and convex on (0, +∞), and let supp g ⊂ If r ≥ 1, then, for any polynomial f ∈ F n , we have Without proof, we note that if the function g in Theorem 8 is not piecewise linear on [0, +∞) with equidistant nodes, then only polynomials of the form f (t) = ce int + νe −int , c, ν ∈ C, are extremal in inequality (7.3) with p ∈ (1, ∞).When p = 1 or p = ∞, a similar conclusion holds, but for the class of trigonometric polynomials of degree at most n.If r > 1, then only polynomials of the form f (t) = ce int + νe −int , c, ν ∈ C, are extremal in inequality (7.4).

Interpolation formulas for periodic functions
If the trigonometric series on the right-hand side of relation (3.2) converges uniformly on T, then one can put the sign of equality in this relation and the obtained equality can be regarded as some interpolation formula.We explain this with the example of the following theorem.
holds for any function f ∈ C(T) such that the series on the left converges uniformly on T.Moreover, µ 0 (n, ψ) + . . .
P r o o f.Consider operator (1.1) for the function ϕ(x) ≡ e iβx ψ(x).Under the conditions of the theorem, we can put the sign of equality in relation (3.2) for ε = 1/n and τ = 1.Therefore, the left-hand side of equality (5.6) can be replaced by the sum of the series in (3.2).We obtain identity (8.1) with accuracy up to the factor e −iβ .The specified properties of the numbers µ k (n, ψ) follow from (5.5) and (5.7).Remark 8. We note that if, for g, we take the function g r (x) = (1 − |x|) r + , r ≥ 1, then, in (8.2), we obtain the interpolation formula of A.I. Kozko [11] (and of M. Riesz and of G. Szegő for r = 1) for the Weyl-Nagy derivative: where µ k (n, g r , β) > 0 for all n ∈ N, k = 0, . . ., 2n − 1, β ∈ R, and r > 1.These coefficients are also positive for r = 1 if n = 1 and β ∈ R or if n ≥ 2 and β = qπ, q ∈ Z.If r = 1, n ≥ 2, and β = πq with q ∈ Z, then, the number of positive coefficients among µ k (n, g 1 , β), k = 0, . . ., 2n − 1, is n + 1 and the remaining are zero (see Remark 7).For r = 1, these coefficients are easily calculated.Since g 1 (t) = 2(1 − cos t)/t 2 , we have Remark 9.It is not difficult to see that all the arguments in the proof of Theorem 9 remain in force also in the case where the 2-periodic continuous function ψ is expanded in an absolutely convergent Fourier series (without the assumption of nonnegativity of the Fourier coefficients c k (ψ)).Therefore, the following statement holds: Assume that a 2-periodic function ψ ∈ C(R) is expanded into an absolutely convergent Fourier series and β ∈ R. Then equality (8.1) holds for any function f ∈ C(T) such that the series on the left in (8.1) converges uniformly on T.
2) Let n ∈ N, and let, for a trigonometric polynomial f ∈ F n , condition (5.9) or (5.10) be satisfied for all integers s = 0, . . ., 2n − 1.Then f (t) = ce int + νe −int , c, ν ∈ C (see Remark 4).The question is, which values of s can be left to have the same conclusion?This is a more general problem than the previous one.
3) To prove or disprove that if, for some s ∈ Z, inequalities µ s (n, ψ) > 0 and µ s+1 (n, ψ) > 0 hold and a function f ∈ C(T) is extremal in inequality (5.8) with p = 1, then f (t) = h(t)g(t), where the function h belongs to L ∞ (T) and has the form (5.4), g ∈ C(T), and g(t) ≥ 0 for t ∈ R.This is true if, in addition, f (t) = 0 for almost all t ∈ R with respect to the Lebesgue measure (see Remark 4 for the case p = 1).
of degree at most n with coefficients in C, where a k := c k + c −k and b k := i(c k − c −k ), k ≥ 0. There are several different definitions of fractional derivative.The following operator for r > 0 and β ∈ R presumably first appeared in the paper by Sz.-Nagy [21, equality (2) for m = 1, λ(k) = k r ].For f ∈ F n , we define f (r,β) (t) := |k|≤n |k| r e iβ sign k c k e ikt = n k=1 k r (a k cos (kt + β) + b k sin (kt + β)) .