Pn (tn )where each and every p j is nonnegative on R (or
Pn (tn )exactly where each and every p j is nonnegative on R (or on R ), j = 1, . . . , n. We recall that p(t) 0 t R p(t) = p2 (t) p2 (t), t R, 2 1 p(t) 0 t R p(t) = p2 (t) tp2 (t), t R 2(4)for some p1 , p2 R[t]. Thus we can express approximating polynomials with regards to sums of squares. This results in characterizations for the existence and uniqueness of the full Markov moment trouble with regards to quadratic forms. The sums of special nonnegative polynomials (4) approximate and dominate . The CLEC-1 Proteins MedChemExpress approximation holds inside the space L1 (Rn ), (respectively in L1 Rn ), where = 1 n , with every single j becoming a moment determinate measure on R (respectively on R ), with finite moments of all orders. We commence with polynomial approximation by nonnegative polynomials on R , for continuous nonnegative functions. The proof of the subsequent outcomes is based on the Stone-Weierstrass theorem plus the properties of partial sums from the alternate Leibniz series obtained from the Taylor series expansion of the Small Ubiquitin Like Modifier 3 Proteins Molecular Weight functions ek (t) = exp(-kt), t [0, ], k N. First, we recall the Kantorovich extension outcome for constructive linear operators [8]. Let X1 be an ordered vector space whose good cone X1, generates X1 (X1 = X1, – X1, ). Recall that in such an ordered vector space X1 , a vector subspace S is known as a majorizing subspace if for any x X1 there exists s S such that x s. The following Kantorovich theorem around the extension of good linear operators holds correct: Theorem 5 (see [8], Theorem 1.two.1). Let X1 be an ordered vector space whose positive cone generates X1 , X0 X1 –a majorizing vector subspace, Y an order total vector space, and T0 : X0 Y a positive linear operator. Then, T0 admits a positive linear extension T : X1 Y . A further extension kind result for linear operators, satisfying a sandwich situation, formulated when it comes to the Markov moment dilemma is stated as follows:Symmetry 2021, 13,six ofTheorem six (see [25], Theorem four). Let X be an ordered vector space, Y an order complete vector lattice, x j j J , y j j J families of components in X and Y, respectively, and T1 , T2 L( X, Y ) two linear operators. The following statements are equivalent: (a) There’s a linear operator T L( X, Y ), such that T1 ( x ) T ( x ) T2 ( x ), x X , T x j = y j , j J; (b) For any finite subset J0 J, and any jj JR, the following implication holds correct:j Jj x j = 2 – 1 ,1 , two Xj Jj y j T2 (2 ) – T1 (1 ).(c)If X is often a vector lattice, then assertions (a) and (b) are equivalent to (c), where (c) T1 (w) T2 (w) for all w X and for any finite subset J0 J and j ; j J0 R, we havej Jj Jj J-.j y j T2 j x j- Tj xjTheorem 6 is usually obtained from a more basic result proved in [26]. Lemma 1 (see [28], Lemma 1). Let : R = [0, ] R be a continuous function, such that lim (t) exists in R . Then there is a decreasing sequence (hl )l in Spanek ; k N, where the functions ek ; k N are defined as follows: ek (t) = exp(-kt), t [0, [ , k N, such that hl (t) (t),t 0, l N = 0, 1, 2, . . ., limhl = uniformly on [0, ). There exists a sequence of polynomial functions ( pl )l N , pl hl , lim pl = , uniformly on compact subsets of R . In particular, such polynomial approximation holds for nonnegative continuous compactly supported functions : R R . In applications, the preceding lemma may very well be useful so as to prove a similar form of result for continuous functions defined only on a compact subset K R , taking values in R . For such a function as : K R , 1 denotes by 0 : R R the extension of , which satisfies 0 (t) = 0 for all t.
