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Polynomial Approximation on Unbounded Subsets, Markov Moment Problem and Other Applications

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Polynomial Approximation on Unbounded Subsets, Markov Moment Problem and Other Applications mathematics Review Polynomial Approximation on Unbounded Subsets, Markov Moment Problem and Other Applications Octav Olteanu Department of Mathematics-Informatics, University Politehnica of Bucharest, 060042 Bucharest, Romania; [email protected] or [email protected] Received: 1 September 2020; Accepted: 23 September 2020; Published: 25 September 2020 Abstract: This paper starts by recalling the author’s results on polynomial approximation over a Cartesian product A of closed unbounded intervals and its applications to solving Markov moment problems. Under natural assumptions, the existence and uniqueness of the solution are deduced. The characterization of the existence of the solution is formulated by two inequalities, one of which involves only quadratic forms. This is the first aim of this work. Characterizing the positivity of a bounded linear operator only by means of quadratic forms is the second aim. From the latter point of view, one solves completely the difficulty arising from the fact that there exist nonnegative polynomials on Rn, n 2, which are not sums of squares. ≥ Keywords: polynomial approximation; Hahn–Banach theorem; Markov moment problem; determinacy; quadratic forms 1. Introduction The present paper refers mainly to various aspects of the classical multidimensional Markov moment problem and its relationship with polynomial approximation on special closed unbounded subsets A of Rn, namely on Cartesian products of closed unbounded intervals. We shall use the notations: j j1 jn n n N = 0, 1, 2, ::: , R+ = [0, ), 'j(t) = t = t ::: t , t = (t1, ::: , tn) A R , j = (j1, ::: , jn) N . f g 1 1 n 2 v 2 The cases A = Rn and, respectively, A = Rn , n 1 will be under attention, since they are + ≥ the most important. The case n = 1 will follow as a consequence, the explicit form of nonnegative polynomials over and, respectively, on being well known (see [1]). The classical real moment R R+ problem can be formulated as follows: find necessary and sufficient conditions on a sequence yj j Nn 2 of real numbers, for the existence of a positive linear functional T L+(E, R), such that 2 n T 'j = yj, j N . (1) 2 n The numbers yj, j N are called the moments with respect to the solution T. Here E is a Banach 2 function space containing both polynomials as well as continuous compactly supported real-valued functions on A. In particular, it results that T can be represented by a positive regular Borel measure on A. In case of a Markov moment problem, one requires also an upper constraint on T such as T P on E, (2) ≤ where P is a continuous convex functional on E. Alternately, a condition of the following type could be imposed T (x) T(x) T (x), x E+ (3) 1 ≤ ≤ 2 2 Mathematics 2020, 8, 1654; doi:10.3390/math8101654 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 1654 2 of 12 where T1, T2 are two given continuous linear functionals on E. Continuity of P, T1, T2 lead to continuity of the solution T of the moment interpolation problem (1) satisfying (2) (respectively, (3)) and, in addition, 1 controls the norm of the solution T. In the sequel, we restrict ourselves to the case when E = Lµ(A), µ being a positive regular Borel moment determinate measure on A. Consider the Markov moment 1 problem of characterizing existence (and eventually uniqueness) of a linear form T : Lµ(A) R which 1 ! satisfies (3), with T1 = 0 and T2 being a given (positive) continuous linear form on Lµ(A). The first step is to define the unique linear form T on the subspace of polynomials such that (1) is verified 0 P 0 1 BX C X B C T0 : R, T0B αj'jC = αj yj P! @B AC j J j J 2 0 2 0 n where J0 N is an arbitrary finite subset and the coefficients αj, j J0 are arbitrary real numbers. ⊂ 1 2 Extending T0 to a linear functional T on Lµ(A) such that 0 T(') T ('), ' L1 (A) (4) ≤ ≤ 2 8 2 µ + is described in Sections2 and3 below. The continuity of T and density of in L1 (A) lead to the P µ uniqueness of the solution T. This method works only for moment determinate measures µ. Recall that a measure µ is called moment determinate (M-determinate) if it is uniquely determinate by its moments Z j n mj = mj(µ) = t dµ, j N . A 2 The first aim of this review paper is to characterize the existence (and uniqueness) of the solution T for the Markov moment problem ((1) and (4)), in terms of quadratic forms. This aim is partially achieved in Section3. Secondly, positivity of a bounded linear operator is completely characterized only by means of quadratic forms. This goal is completely attained. The interested reader can find more information on the moment problem in the monographs [1–4]. Basic results in measure theory, Banach lattices, order complete Banach lattices, related examples and inequalities, and new aspects of the Hahn–Banach theorem can be found in [5–9]. This paper is directly related to the references [4,10–16] and indirectly linked with the works [5–9,17–38]. The interested reader can find connections of the moment problem with other fields of analysis (other than polynomial approximation), such as operator theory, fixed point theory, optimization, and inverse problems. At the end of the Introduction, it is worth noticing that all our results are valid not only for real valued linear (respectively convex continuous) functionals, but also for operators having as codomain an order complete Banach lattice F. The order completeness assumption allows applying the Hahn–Banach result stated in Section2 (Theorem 1 below). Other recent results on the Hahn–Banach theorem, sandwich theorem and/or their applications have been published in [39–42]. The rest of the paper is organized as follows. Section2 reviews the methods used along this work, specifying some of the corresponding references and recalling a well-known extension result for linear positive operators, which holds preserving linearity and positivity. In Section3, the main results specified in the Abstract are stated and partially proved. Section4 concludes the paper. 2. Materials and Methods The basic methods applied along this work are: (1) Applying the next result, Theorem 1 stated below, sometimes called Lemma of the majorizing subspace (see [3–5,11–16,21,22,24] for the proof or/and related applications), accompanied by polynomial approximation, to prove the existence of a positive solution T. Let E1 be an ordered Mathematics 2020, 8, 1654 3 of 12 vector spaces for which the positive cone E + is generating (E = E + E +). Recall that in 1, 1 1, − 1, such an ordered vector space E1, a vector subspace S is called a majorizing subspace if for any x E , there exists s S such that x s. 2 1 2 ≤ Theorem 1. Let E be an ordered vector space whose positive cone is generating, S E a majorizing vector 1 ⊂ 1 subspace, F an order complete vector space, T : S F a positive linear operator. Then T admits a linear 0 ! 0 positive extension T : E F. 1 ! Using other variants of the Hahn–Banach theorem (separation theorem). For more general Hahn–Banach type theorems see [17–20]. n Characterizing the existence of the solution T in terms of its moments yj, j N . 2 (2) Measure theory detailed results discussed in [7]. (3) Polynomial approximation on unbounded subsets, (see [4,10–16]) recalled and applied in Section3. n Characterizing the existence of the solution T in terms of its moments yj, j N and quadratic 2 forms. Uniqueness of the solution follows too. Applying the notion of a moment-determinate measure. Establishing determinacy or indeterminacy of a measure requires special criterions (see [3,29,30]). (4) Evaluating the norm of the solution T in terms of the norm of the given continuous linear functional (or operator) T2, under conditions (4). This goal is achieved in theorems of Section3 as well. 3. Results 3.1. Solving Markov Moment Problem Over Unbounded Subsets via Polynomial Approximation Polynomial approximation recalled below allow proving the existence as well as the uniqueness of the solution of some classical moment problems on unbounded subsets. Lemma 1. Let : [0, ) R+ be a continuous function, such that lim (t) R+ exists. Then there is a 1 ! t 2 !1 decreasing sequence (hl)l in the linear hull of the functions 'k(t) = exp( kt), k N, t [0, ), − 2 2 1 such that h (t) > (t), t 0, l N, limh = uniformly on [0, ). There exists a sequence of polynomial l ≥ 2 l 1 ( ) = [ ) functions epl l N, epl hl > , lim epl , uniformly on compact subsets of 0, . 2 ≥ 1 Lemma 2. Let ν be an M-determinate positive regular measure on [0, ), with finite moments of all natural 1 1 orders. If , (epl) are as in Lemma 1, then there exists a subsequence (eplm) , such that epl in L ([0. )) l m m ! ν 1 and uniformly on compact subsets. In particular, it follows that the positive cone + of nonnegative polynomials P is dense in the positive cone L1([0. )) of L1([0. )). ν 1 + ν 1 Lemma 3. Let A Rn be an unbounded closed subset, and ν an M-determinate positive regular Borel measure ⊂ on A, with finite moments of all natural orders. Then for any x C0(A), x(t) 0, t A, there exists a 1 2 ≥ 8 2 sequence (pm) , pm P+, pm x, pm x in L (A). In particular, we have m 2 ≥ ! ν Z Z lim pm(t)dν = x(t)dν, A A 1 1 + is dense in L (A) , and is dense in L (A).
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