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 Moveltipril Autophagy denote the set of all measurable function f defined in (-1, 1) such that the 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’ll create k y (or k x ) to be able to regard k as the univariate function inside the only variable x (or y). two.2. 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) can be rewritten inside the following type:( I – K ) f = g,GSK2646264 Autophagy exactly where I denotes the identity operator. In an effort to give the enough conditions assuring the compactness from the operator K : Cu Cu , we must recall the following definition. For any f Cu and with 0 r N, in [10], it was defined the following modulus of smoothness: r ( f , t)u = sup exactly where Ihr = [-1 + 4h2 r2 , 1 – 4h2 r2 ] and also the following will be 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 as an example [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 from the Equation (1) in the space Cu and for which its proof is provided in Section 6.Mathematics 2021, 9,four ofTheorem 1. Beneath 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,(4)the operator K : Cu Cu is compact. For that reason, if ker( I – K ) = 0, for any g Cu , Equation (1) admits a exceptional resolution in Cu . Remark 1. We observe that (4) is happy also when the kernel k( x, y) in (3) is weakly singular. For instance k( x, y) = | x – y|, -1 0 , fulfils the assumption with s = 1 + (see ([11], Lemma four.1, p. 322) and ([3], pp. three)). 2.three. Product Integration Guidelines Denoted by pm (w)mN , the method of your orthonormal polynomials with respect towards the Jacobi weight w = v, , , -1, the polynomial pm (w) is so defined: pm (w, x ) = m (w) x m + terms of reduced 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 solution 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)In line with a consolidated terminology, we are going to refer to the solution integration rule in (5) as Ordinary Item Rule only to distinguish it in the extended solution integration rule introduced under. Moreover, we recall that Mi (y)iN are referred to as Modified Moments [12] (see, e.g., [13]). With respect to the stability and the convergence with the preceding rule, the following result, helpful for our aim, can be deduced by ([9], p. 348) (see also [14]). Theorem two. 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 ). As well as the prior well-known item rule, we recall the following Extended Solution Rule (see [8]) based on the zeros of pm (w) pm+1 (w). Denoted by {yk := xm+1,k.
