How that f ( is not Doss-( p(, c)-almost Spautin-1 custom synthesis periodic if c C \ -1, 1. This really is clear for c = 0; for the remainder, it suffices to show that f ( will not be Doss-(1, c)-almost periodic if c C \ -1, 0, 1. Suppose that c = rei for some r 0 and (-, ]. Suppose further that cos = ; then r = 1 and sin = 0 so thatt-tcos(s ) – c cos sptds =-tcos(s ) – r cos spds,for any R. Since the function s | cos(s ) – r cos s|, s R is periodic and not identically equal to zero, the final estimate yields the existence of a finite genuine number c 0 such that ( R is provided in advance):t-tcos(s ) – c cos spds c (t/2) ,t 0,which merely yields a contradiction. Therefore, cos = and there exists a constant d (0, 1) such that, for every single R, we’ve got:t-tcos(s ) – c cos stpdsp=-tcos2 (s ) – 2r cos cos s cos(s ) r2 cos2 s dst(dr)1/p-tcos s cos(s )pds.Since the function s | cos s cos(s )|, s R is periodic and not identically equal to zero, the last estimate yields the existence of a finite actual number d 0 such that ( R is offered in advance):Mathematics 2021, 9,12 oft-tcos(s ) – c cos spds d (cr)1/p (t/2) ,t 0,which implies the essential. Example 6 (cf. also [12] (Example 2.22), and [12] (Instance 2.23) for the pointwise products of c-almost periodic functions). Suppose that c S1 and p( D (R). Then we’ve the following: (i) Suppose that c = 1. Then the space consisting of all Doss-p(-uniformly recurrent Cytochalasin B site functions isn’t a vector space with all the usual operations as simply shown. Now we’ll prove that the space of Doss-1-almost periodic functions can also be not a vector space with the usual operations. Define f : R R by f := 0 for x 0, f ( x) := n/2 if x (n – 2, n – 1] for some x n 2N and f ( x) := – n/2 if x (n – 1, n] for some n 2N. Then we understand that the function f ( is Weyl-1-almost periodic too as that for each and every n 2N we have 1 sup l 2l tR liml-lf (t n x) – f (t x) dx = 0,(six)and that for every actual number 2Z we’ve / 1 l 2l limlf ( x ) – f ( x) dx = ;(7)see J. Stryja [29] (pp. 427), [11] (Instance four.28) and [7] (Instance 8.3.20). Define g : R R by g(t) := cos t, t R. Hence, the functions f ( and g( are Doss-1-almost periodic (cf. also Section 2.1 beneath); but, its sum is not Doss-1-almost periodic. In actual truth, if 2Z, / then the consideration from the above instance along with the equation (7) indicates that there exists a finite true number d 0 such that 1 ll| f ( x ) cos( x ) – f ( x) – cos x | dx1 l | cos( x ) – cos x | dx l 0 0 l | sin(/2)| l | f ( x ) – f ( x)| dx – | sin( x (/2))| dx l 0l1 l 1 = l 1 l| f ( x ) – f ( x)| dx -l| f ( x ) – f ( x)| dx – d ,l .If 2Z, then the Equation (6) yields that there exist two finite genuine numbers d 0 and l0 0 such that 1 ll| f ( x ) cos( x ) – f ( x) – cos x | dx1 l 1 l | cos( x ) – cos x | dx – | f ( x ) – f ( x)| dx l 0 l 0 l 1 1 l | sin(/2)| | sin( x (/2))| dx – | f ( x ) – f ( x)| dx 2l l 0 0 1 l d- | f ( x ) – f ( x)| dx d/2, l l0 . lThis implies the required statement; observe also that the above evaluation implies that the collection of all Weyl-1-almost periodic functions has not a linear vector structure with the usual operations. We deeply think that the collection of all Doss-p(-almost periodic functions as well as the collection of all Weyl-p(-almost periodic functions usually are not vector spaces with the usual operations, too.Mathematics 2021, 9,13 of(ii)Suppose that c = -1. Then the space consisting of all Doss-( p(, c)-almost periodic functions isn’t a vector space with the usual operations since the functions 2-1 cos(.