On the Sets of Convergence for Sequences of the 𝑞-Bernstein Polynomials with 𝑞>1

Table 1

The values of 𝐸(𝑛,𝑞,𝑥)=𝐵_𝑛,𝑞(𝑓;𝑥)−𝑓(𝑥) at some points 𝑥∈[0,1].

𝑥	𝐸(3,2,𝑥)	𝐸(4,2,𝑥)	𝐸(10,2,𝑥)	𝐸(15,2,𝑥)	𝐸(20,2,𝑥)	𝐸(25,2,𝑥)	𝐸(30,2,𝑥)
(𝑞+1)/2𝑞	−8.76×10^−2	0.336	7.07×10^9	−6.08×10^26	1.76×10^51	−1.71×10^83	5.55×10^122
1/𝑞	−9.00×10^−3	−4.02×10^−3	−5.70×10^−5	−1.78×10^−6	−5.56×10^−8	−1.74×10^−9	−5.43×10^−11
(𝑞+1)/2𝑞^2	9.88×10^−3	−4.63×10^−2	−1.15×10^7	3.09×10^22	−2.79×10^45	8.47×10^75	−8.62×10^113
1/𝑞^2	−2.39×10^−2	−9.99×10^−3	−1.36×10^−4	−4.23×10^−6	−1.32×10^−7	−4.13×10^−9	−1.29×10^−10
(𝑞+1)/2𝑞^3	−6.66×10^−2	1.39×10^−3	4.90×10^4	−4.09×10^18	1.16×10^40	−1.09×10^69	3.48×10^105
1/𝑞^3	−0.126	−2.05×10^−2	−2.52×10^−4	−7.84×10^−6	−2.45×10^−7	−7.66×10^−9	−2.39×10^−10
(𝑞+1)/2𝑞^4	−0.159	−5.06×10^−2	−465.	1.21×10^15	−1.06×10^35	3.15×10^62	−3.13×10^97
1/𝑞^4	−0.190	−9.47×10^−2	−4.29×10^−4	−1.33×10^−5	−4.17×10^−7	−1.30×10^−8	−4.07×10^−10
(𝑞+1)/2𝑞^5	−0.201	−0.12	9.37	−7.49×10^11	2.06×10^30	−1.91×10^56	5.93×10^89
1/𝑞^5	−0.206	−0.145	−7.07×10^−4	−2.19×10^−5	−6.84×10^−7	−2.14×10^−8	−6.68×10^−10
(𝑞+1)/2𝑞^6	−0.204	−0.155	−0.393	9.54×10^8	−8.21×10^25	2.37×10^50	−2.30×10^82
1/𝑞^6	−0.194	−0.16	−1.15×10^−3	−3.53×10^−5	−1.10×10^−6	−3.45×10^−8	−1.08×10^−9
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
1/𝑞^19	−1.24×10^−2	−1.24×10^−2	−1.22×10^−2	−1.05×10^−2	−5.43×10^−4	−1.43×10^−5	−4.45×10^−7
(𝑞+1)/2𝑞^20	−1.13×10^−2	−1.13×10^−2	−1.11×10^−2	−9.82×10^−3	−1.21×10^−3	1.27	−2.23×10^10
1/𝑞^20	−9.84×10^−3	−9.84×10^−3	−9.75×10^−3	−8.87×10^−3	−2.24×10^−3	−2.28×10^−5	−7.06×10^−7
(𝑞+1)/2𝑞^21	−8.94×10^−3	−8.94×10^−3	−8.87×10^−3	−8.22×10^−3	−2.86×10^−3	−0.102	2.91×10^7
1/𝑞^21	−7.81×10^−3	−7.81×10^−3	−7.76×10^−3	−7.33×10^−3	−3.49×10^−3	−5.51×10^−2	−1.12×10^−6
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
1/𝑞^49	−1.21×10^−5	−1.21×10^−5	−1.21×10^−5	−1.21×10^−5	−1.21×10^−5	−1.21×10^−5	−1.21×10^−5
(𝑞+1)/2𝑞^50	−1.10×10^−5	−1.10×10^−5	−1.10×10^−5	−1.10×10^−5	−1.10×10^−5	−1.10×10^−5	−1.10×10^−5
1/𝑞^50	−9.61×10^−6	−9.61×10^−6	−9.61×10^−6	−9.61×10^−6	−9.61×10^−6	−9.61×10^−6	−9.61×10^−6