Norm: fWr (u):= fCu+ f (r ) r u.Finally, by LNorm: fWr (u):= fCu+

Norm: fWr (u):= fCu+ f (r ) r u.Finally, by L
Norm: fWr (u):= fCu+ f (r ) r u.Lastly, by L log+ L, we denote the set of all measurable function f defined in (-1, 1) such that the CFT8634 Epigenetics following is definitely the case:-| f ( x )|(1 + log+ | f ( x )|)dx ,log+ f ( x ) = log(max(1, f ( x ))).For any bivariate function k( x, y), we will write k y (or k x ) so that you can regard k because the univariate function inside the only variable x (or y). two.two. Solvability with the Equation (1) in Cu Let us set the following:(K f )(y) = -f ( x )k( x, y)( x ) dx,= v, ,, -1.(three)Equation (1) could be rewritten within the following form:( I – K ) f = g,where I denotes the identity operator. As a way to give the enough situations assuring the compactness on the operator K : Cu Cu , we need to recall the following definition. For any f Cu and with 0 r N, in [10], it was defined the following SBP-3264 site modulus of smoothness: r ( f , t)u = sup where Ihr = [-1 + 4h2 r2 , 1 – 4h2 r2 ] and also the following is definitely the case: r f ( x ) = h0 h t( r f ) u hIhr,i =(-1)irr h f x + (r – 2i )h( x ) . iFor any f Wr (u), the modulus r ( f , t)u is estimated by implies on the following inequality (see for instance [11], p. 314): r ( f , t)u C sup hr f (r) r u0 h t Ihr,C = C( f , t).We’re now capable to state a Theorem that guarantees the solvability of your Equation (1) in the space Cu and for which its proof is offered in Section six.Mathematics 2021, 9,four ofTheorem 1. Under the following assumptions, with 0 s r and C = C( f ), ky L1 ([-1, 1]), u |y|1 sup supt r ( K f , t ) u tsC fCu,(four)the operator K : Cu Cu is compact. Consequently, if ker( I – K ) = 0, for any g Cu , Equation (1) admits a distinctive resolution in Cu . Remark 1. We observe that (4) is happy also when the kernel k( x, y) in (3) is weakly singular. As an example k( x, y) = | x – y|, -1 0 , fulfils the assumption with s = 1 + (see ([11], Lemma 4.1, p. 322) and ([3], pp. three)). two.three. Solution Integration Guidelines Denoted by pm (w)mN , the technique from the orthonormal polynomials with respect to the Jacobi weight w = v, , , -1, the polynomial pm (w) is so defined: pm (w, x ) = m (w) x m + terms of lower degree, m (w) 0.Let xk := xm,k (w) : k = 1, . . . , m be the zeros of pm (w) and let the following:m –m,k := m,k (w) =i =p2 (w, xk ) i,k = 1, . . . , m,be the Christoffel numbers with respect to w. For the following integral:I( f , y) =-f ( x )k( x, y)( x ) dxconsider the following item integration rule:I( f , y) =wherek =I I Ck (y) f (xk ) + em ( f , y) =: Im ( f , y) + em ( f , y),m(five)m- Ck (y) = m,k i=01 pi (w, xk ) Mi (y),Mi ( y ) =1 -pi (w, x )k( x, y)( x ) dx, i = 0, 1, . . . , m – 1. (6)According to a consolidated terminology, we will refer for the item integration rule in (5) as Ordinary Product Rule only to distinguish it in the extended product integration rule introduced below. Furthermore, we recall that Mi (y)iN are generally known as Modified Moments [12] (see, e.g., [13]). With respect to the stability and the convergence of the previous rule, the following outcome, beneficial for our aim, is usually deduced by ([9], p. 348) (see also [14]). Theorem 2. Beneath the following assumptions: ky sup L log+ L, u |y|1 ky sup L1 ([-1, 1]), w |y|w L1 ([-1, 1]), u(7)for any f Cu , we get the following bounds: sup sup |Im ( f , y)| C fm|y|CuandI sup em ( f , y) C Em-1 ( f )u , |y|Mathematics 2021, 9,five ofwith C = C(m, f ). In addition to the previous well-known solution rule, we recall the following Extended Item Rule (see [8]) depending on the zeros of pm (w) pm+1 (w). Denoted by {yk := xm+1,k.