New modification of the Post Widder operators preserving exponential functions

Figures & data

Figure 1. About ${\sigma }_{\lambda }$ and ${\rho }_{\lambda }.$

Figure 2. Behaviour of $P_{\lambda }^{\theta }$ operators for various values of λ and μ.

Figure 3. Comparison of operators $P_{\lambda }^{\theta }(f;x)$ and $P_{\lambda ,\alpha }^{*}(f;x)$ for different λ values.

Table 1. The values of ${\sigma }_{\lambda }$ for various λ and μ in the interval $x\in [0.1,20].$

λ	$\mu =0.25$	$\mu =0.5$	$\mu =0.75$	μ =  1	$\mu =1.25$
50	0.0101506913953	0.0164649246670	0.0244687401054	0.034373606654	0.114654279
75	0.0067620928581	0.0109199853118	0.0161298883207	0.022484666940	0.030102290
100	0.0050695332037	0.0081681916317	0.0120276616287	0.016699839271	0.022250028
125	0.0040546102332	0.0065239445991	0.0095883173189	0.013280805769	0.017642789

Table 2. The values of ${\rho }_{\lambda }$ for various λ and μ in the interval $x\in [0.1,20].$

λ	$\mu =0.25$	$\mu =0.5$	$\mu =0.75$	μ =  1	$\mu =1.25$
50	0.0075337890598	0.0101015913706	0.0130553239149	0.0164151540271	0.0202047397
75	0.0050244938197	0.0067247763708	0.0086713306489	0.0108731469581	0.0133407054
100	0.0037690406841	0.0050397960288	0.0064910512850	0.0081278703462	0.0099561368
125	0.0030155365685	0.0040299724812	0.0051867714986	0.0064891769777	0.0079409496

Table 3. Comparison Of the errors for $P_{\lambda ,\alpha }^{*},a=2$ and $P_{\lambda }^{\theta }$ with respect to various values of $\mu \in \{0.25,0.5,1\},$ for function $f(x)=e^{-5x}$ in $x\in [0.1,4].$

λ	$P_{\lambda ,\alpha }^{*},a=2$	$P_{\lambda }^{\theta },\mu =0.25$	$P_{\lambda }^{\theta },\mu =0.5$	$P_{\lambda }^{\theta },\mu =1$
10	0.0211165837883	0.0104160822281	0.0128387920967	0.0193125431808
25	0.0060255257096	0.0034631541717	0.0040218632162	0.0055365258294
50	0.0027102082215	0.0017414012825	0.0019895931130	0.0025308291489
75	0.0018110450521	0.0011631367734	0.0013290402874	0.0016908893096
100	0.0013598799550	0.0008731813121	0.0009977755384	0.0012695484868
125	0.0010886717521	0.0006989435513	0.0007986990840	0.0010163032756

Figure 4. Comparison of operators $P_{\lambda }^{\theta }(f;x)$ and $P_{\lambda ,\alpha }^{*}(f;x)$ for different λ values.

Table 4. Comparison of the errors for $P_{\lambda ,\alpha }^{*},a=2$ and $P_{\lambda }^{\theta }$ with respect to various values of $\mu \in \{0.25,0.5,0.6\},$ for function $f(x)=(2x^4+3x^2+6)e^{-2x}$ in $x\in [0.1,5].$

λ	$P_{m,\alpha }^{*},a=2$	$P_{\lambda }^{\theta },\mu =0.25$	$P_{\lambda }^{\theta },\mu =0.5$	$P_{\lambda }^{\theta },\mu =0.6$
10	0.3580078363242	0.1636569640768	0.2428086366319	0.2812895777007
25	0.1356249060185	0.0645460265532	0.0944989489807	0.1090011457796
50	0.0660639319941	0.0320722662244	0.0466945717810	0.0537644561336
75	0.0436121407380	0.0213329668881	0.0309963808286	0.0356665216378
100	0.0325414463369	0.0159809493762	0.0231958027528	0.0266818157290
125	0.0259510498680	0.0127755988459	0.0185315429465	0.0213122775669