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Integral Baskakov type operator with quadratic order of approximation

  • *Corresponding author: Asha Ram Gairola

    *Corresponding author: Asha Ram Gairola 
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  • In order to approximate Lebesgue integrable functions on the positive real axis, we propose an integral variant of Baskakov operators. We show that the linear operators so obtained exhibit the quadratic approximation order $ O(n^{-2}) $ for sufficiently large values of $ n. $

    Mathematics Subject Classification: Primary: 41A36, 41A25; Secondary: 41A81.

    Citation:

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  • Figure 1.  $ f(x) $ versus $ \widehat{S}_n(f;x), $ $ n = 5, 10 $

    Figure 2.  Absolute error function $ E_n(x) = |f(x)-\widehat{S}_n(f;x)|, $ $ n = 5, 10 $

    Figure 3.  $ \frac{x}{(x+1)^2} $ versus $ \widehat{S}_n(f;x), $ $ n = 5, 15, 25 $

    Figure 4.  Absolute error function $ E_n(x), n = 5, 15, 25 $

    Table 1.  $ f(x) $ versus $ \widehat{S}_n(f;x), $ $ n = 5, 10 $

    $ \widehat{S}_n(f;x) $ $ |f(x)-\widehat{S}_n(f;x)| $
    $ x $ $ f(x) $ $ n=5 $ $ n=10 $ $ n=5 $ $ n=10 $
    0 0.04 0 0 0.04 0.04
    5 0.0246154 0.0299839 0.00975022 0.00536847 0.0148652
    10 0.0080798 0.0107168 -0.0051111 0.00263701 0.0131909
    15 0.00386969 -0.00449091 -0.00126303 0.0083606 0.00513272
    20 0.0022559 -0.00872948 -0.000310729 0.0109854 0.00256663
    25 0.00147447 -0.00877652 -8.91634E-05 0.010251 0.00156363
    30 0.00103824 -0.00765977 -2.96435E-05 0.00869801 0.00106788
    35 0.000770333 -0.0063892 -1.11478E-05 0.00715953 0.000781481
    40 0.000594135 -0.00525737 -4.63868*$ 10^{-6} $ 0.00585151 0.000598774
     | Show Table
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    Table 2.  $ f(x) $ with respect to $ \widehat{S}_n(f;x), $ $ n = 5, 15, 25 $

    $ \widehat{S}_n(f;x) $ $ |f(x)-\widehat{S}_n(f;x)| $
    $ x $ $ f(x) $ $ n=5 $ $ n=15 $ $ n=25 $ $ n=5 $ $ n=15 $ $ n=25 $
    0 0 0 0 0 0 0 0
    10 0.0826446 0.0799523 0.0785445 0.00410016 0.00269229 0.00410016 0.00073308
    20 0.0453515 0.0242172 0.0402563 0.00509522 0.0211342 0.00509522 0.000690846
    30 0.0312175 0.00791587 0.026823 0.00439447 0.0233016 0.00439447 0.000555392
    40 0.0237954 0.00147329 0.0200633 0.00373209 0.0223221 0.00373209 0.000456052
    50 0.0192234 -0.00148503 0.01601 0.00321334 0.0207084 0.00321334 0.000384908
    60 0.0161247 -0.00295178 0.0133136 0.00281113 0.0190765 0.00281113 0.000332307
    70 0.0138861 -0.00369866 0.0113919 0.00249423 0.0175848 0.00249423 0.00029208
    80 0.0121933 -0.00406924 0.00995373 0.00223953 0.0162625 0.00223953 0.00026041
    90 0.0108683 -0.00423216 0.00883727 0.00203098 0.0151004 0.00203098 0.000234868
    100 0.00980296 -0.00427638 0.0079456 0.00185736 0.0140793 0.00185736 0.00021385
     | Show Table
    DownLoad: CSV
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