Namespaces
Variants
Actions

Difference between revisions of "Baskakov operators"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
V.A. Baskakov [[#References|[a2]]] introduced a sequence of linear positive operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b1101501.png" /> with weights
+
<!--
 +
b1101501.png
 +
$#A+1 = 28 n = 1
 +
$#C+1 = 28 : ~/encyclopedia/old_files/data/B110/B.1100150 Baskakov operators
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b1101502.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
V.A. Baskakov [[#References|[a2]]] introduced a sequence of linear positive operators  $  L _ {n} $
 +
with weights
 +
 
 +
$$
 +
p _ {nk }  ( x ) = ( - 1 )  ^ {k} {
 +
\frac{x  ^ {k} }{k! }
 +
} \phi _ {n} ^ {( k ) } ( x )
 +
$$
  
 
by
 
by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b1101503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
( L _ {n} f ) ( x ) = \sum _ {k = 0 } ^  \infty  p _ {nk }  ( x ) f \left ( {
 +
\frac{k}{n}
 +
} \right ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b1101504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b1101505.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b1101506.png" />, for all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b1101507.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b1101508.png" /> for which the series converges. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b1101509.png" /> is a sequence of functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015010.png" /> having the following properties for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015012.png" />:
+
where $  n \in \mathbf N $,
 +
$  x \in [ 0,b ] $,  
 +
b > 0 $,  
 +
for all functions $  f $
 +
on $  [ 0, \infty ) $
 +
for which the series converges. Here, $  \{ \phi _ {n} \} _ {n \in \mathbf N }  $
 +
is a sequence of functions defined on $  [0,b] $
 +
having the following properties for every $  n,k \in \mathbf N $,
 +
$  k > 0 $:
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015013.png" />;
+
i) $  \phi _ {n} \in C  ^  \infty  [ 0,b ] $;
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015014.png" />;
+
ii) $  \phi _ {n} ( 0 ) = 1 $;
  
iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015015.png" /> is completely monotone, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015016.png" />;
+
iii) $  \phi _ {n} $
 +
is completely monotone, i.e., $  ( - 1 )  ^ {k} \phi _ {n} ^ {( k ) } \geq  0 $;
  
iv) there exists an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015017.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015019.png" />.
+
iv) there exists an integer $  c $
 +
such that $  \phi _ {n} ^ {( k+1 ) } = - n \phi _ {n + c }  ^ {( k ) } $,
 +
$  n >  \max  \{ 0, - c \} $.
  
 
Baskakov studied convergence theorems of bounded continuous functions for the operators (a1). For saturation classes for continuous functions with compact support, see [[#References|[a8]]]. For a result concerning bounded continuous functions, see [[#References|[a3]]].
 
Baskakov studied convergence theorems of bounded continuous functions for the operators (a1). For saturation classes for continuous functions with compact support, see [[#References|[a8]]]. For a result concerning bounded continuous functions, see [[#References|[a3]]].
  
In his work on Baskakov operators, C.P. May [[#References|[a6]]] took conditions slightly different from those mentioned above and showed that the local inverse and saturation theorems hold for functions with growth less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015020.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015021.png" />. Bernstein polynomials and Szász–Mirakian operators are the particular cases of Baskakov operators considered by May.
+
In his work on Baskakov operators, C.P. May [[#References|[a6]]] took conditions slightly different from those mentioned above and showed that the local inverse and saturation theorems hold for functions with growth less than $  ( 1 + t )  ^ {N} $
 +
for some $  N > 0 $.  
 +
Bernstein polynomials and Szász–Mirakian operators are the particular cases of Baskakov operators considered by May.
  
 
S.P. Singh [[#References|[a7]]] studied simultaneous approximation, using another modification of the conditions in the original definition of Baskakov operators. However, it was shown that his result is not correct (cf., e.g., [[#References|[a1]]], Remarks).
 
S.P. Singh [[#References|[a7]]] studied simultaneous approximation, using another modification of the conditions in the original definition of Baskakov operators. However, it was shown that his result is not correct (cf., e.g., [[#References|[a1]]], Remarks).
  
Motivated by the Durrmeyer integral modification of the Bernstein polynomials, M. Heilmann [[#References|[a4]]] modified the Baskakov operators in a similar manner by replacing the discrete values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015022.png" /> in (a1) by an integral over the weighted function, namely,
+
Motivated by the Durrmeyer integral modification of the Bernstein polynomials, M. Heilmann [[#References|[a4]]] modified the Baskakov operators in a similar manner by replacing the discrete values $  f ( {k / n } ) $
 +
in (a1) by an integral over the weighted function, namely,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015023.png" /></td> </tr></table>
+
$$
 +
( M _ {n} f ) ( x ) = \sum _ {k = 0 } ^  \infty  p _ {nk }  ( x ) ( n - c ) \int\limits _ { 0 } ^  \infty  {p _ {nk }  ( t ) f ( t ) }  {dt } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015024.png" /></td> </tr></table>
+
$$
 +
n > c,  x \in [ 0, \infty ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015025.png" /> is a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015026.png" /> for which the right-hand side is defined. He studied global direct and inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015027.png" />-approximation theorems for these operators.
+
where $  f $
 +
is a function on $  [ 0, \infty ) $
 +
for which the right-hand side is defined. He studied global direct and inverse $  L _ {p} $-
 +
approximation theorems for these operators.
  
Subsequently, a global direct result for simultaneous approximation in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015028.png" />-metric in terms of the second-order Ditzian–Totik modulus of smoothness was proved, see [[#References|[a5]]]. For local direct results for simultaneous approximation of functions with polynomial growth, see [[#References|[a5]]].
+
Subsequently, a global direct result for simultaneous approximation in the $  L _ {p} $-
 +
metric in terms of the second-order Ditzian–Totik modulus of smoothness was proved, see [[#References|[a5]]]. For local direct results for simultaneous approximation of functions with polynomial growth, see [[#References|[a5]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.N. Agrawal,  H.S. Kasana,  "On simultaneous approximation by Szász–Mirakian operators"  ''Bull. Inst. Math. Acad. Sinica'' , '''22'''  (1994)  pp. 181–188</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.A. Baskakov,  "An example of a sequence of linear positive operators in the space of continuous functions"  ''Dokl. Akad. Nauk SSSR'' , '''113'''  (1957)  pp. 249–251  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Berens,  "Pointwise saturation of positive operators"  ''J. Approx. Th.'' , '''6'''  (1972)  pp. 135–146</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Heilmann,  "Approximation auf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015029.png" /> durch das Verfahren der Operatoren vom Baskakov–Durrmeyer Typ" , Univ. Dortmund  (1987)  (Dissertation)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Heilmann,  M.W. Müller,  "On simultaneous approximation by the method of Baskakov–Durrmeyer operators"  ''Numer. Funct. Anal. Optim.'' , '''10'''  (1989)  pp. 127–138</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C.P. May,  "Saturation and inverse theorems for combinations of a class of exponential-type operators"  ''Canad. J. Math.'' , '''28'''  (1976)  pp. 1224–1250</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S.P. Singh,  "On Baskakov-type operators"  ''Comment. Math. Univ. St. Pauli,'' , '''31'''  (1982)  pp. 137–142</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  Y. Suzuki,  "Saturation of local approximation by linear positive operators of Bernstein type"  ''Tôhoku Math. J.'' , '''19'''  (1967)  pp. 429–453</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.N. Agrawal,  H.S. Kasana,  "On simultaneous approximation by Szász–Mirakian operators"  ''Bull. Inst. Math. Acad. Sinica'' , '''22'''  (1994)  pp. 181–188</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.A. Baskakov,  "An example of a sequence of linear positive operators in the space of continuous functions"  ''Dokl. Akad. Nauk SSSR'' , '''113'''  (1957)  pp. 249–251  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Berens,  "Pointwise saturation of positive operators"  ''J. Approx. Th.'' , '''6'''  (1972)  pp. 135–146</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Heilmann,  "Approximation auf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110150/b11015029.png" /> durch das Verfahren der Operatoren vom Baskakov–Durrmeyer Typ" , Univ. Dortmund  (1987)  (Dissertation)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Heilmann,  M.W. Müller,  "On simultaneous approximation by the method of Baskakov–Durrmeyer operators"  ''Numer. Funct. Anal. Optim.'' , '''10'''  (1989)  pp. 127–138</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C.P. May,  "Saturation and inverse theorems for combinations of a class of exponential-type operators"  ''Canad. J. Math.'' , '''28'''  (1976)  pp. 1224–1250</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S.P. Singh,  "On Baskakov-type operators"  ''Comment. Math. Univ. St. Pauli,'' , '''31'''  (1982)  pp. 137–142</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  Y. Suzuki,  "Saturation of local approximation by linear positive operators of Bernstein type"  ''Tôhoku Math. J.'' , '''19'''  (1967)  pp. 429–453</TD></TR></table>

Revision as of 10:13, 29 May 2020


V.A. Baskakov [a2] introduced a sequence of linear positive operators $ L _ {n} $ with weights

$$ p _ {nk } ( x ) = ( - 1 ) ^ {k} { \frac{x ^ {k} }{k! } } \phi _ {n} ^ {( k ) } ( x ) $$

by

$$ \tag{a1 } ( L _ {n} f ) ( x ) = \sum _ {k = 0 } ^ \infty p _ {nk } ( x ) f \left ( { \frac{k}{n} } \right ) , $$

where $ n \in \mathbf N $, $ x \in [ 0,b ] $, $ b > 0 $, for all functions $ f $ on $ [ 0, \infty ) $ for which the series converges. Here, $ \{ \phi _ {n} \} _ {n \in \mathbf N } $ is a sequence of functions defined on $ [0,b] $ having the following properties for every $ n,k \in \mathbf N $, $ k > 0 $:

i) $ \phi _ {n} \in C ^ \infty [ 0,b ] $;

ii) $ \phi _ {n} ( 0 ) = 1 $;

iii) $ \phi _ {n} $ is completely monotone, i.e., $ ( - 1 ) ^ {k} \phi _ {n} ^ {( k ) } \geq 0 $;

iv) there exists an integer $ c $ such that $ \phi _ {n} ^ {( k+1 ) } = - n \phi _ {n + c } ^ {( k ) } $, $ n > \max \{ 0, - c \} $.

Baskakov studied convergence theorems of bounded continuous functions for the operators (a1). For saturation classes for continuous functions with compact support, see [a8]. For a result concerning bounded continuous functions, see [a3].

In his work on Baskakov operators, C.P. May [a6] took conditions slightly different from those mentioned above and showed that the local inverse and saturation theorems hold for functions with growth less than $ ( 1 + t ) ^ {N} $ for some $ N > 0 $. Bernstein polynomials and Szász–Mirakian operators are the particular cases of Baskakov operators considered by May.

S.P. Singh [a7] studied simultaneous approximation, using another modification of the conditions in the original definition of Baskakov operators. However, it was shown that his result is not correct (cf., e.g., [a1], Remarks).

Motivated by the Durrmeyer integral modification of the Bernstein polynomials, M. Heilmann [a4] modified the Baskakov operators in a similar manner by replacing the discrete values $ f ( {k / n } ) $ in (a1) by an integral over the weighted function, namely,

$$ ( M _ {n} f ) ( x ) = \sum _ {k = 0 } ^ \infty p _ {nk } ( x ) ( n - c ) \int\limits _ { 0 } ^ \infty {p _ {nk } ( t ) f ( t ) } {dt } , $$

$$ n > c, x \in [ 0, \infty ) , $$

where $ f $ is a function on $ [ 0, \infty ) $ for which the right-hand side is defined. He studied global direct and inverse $ L _ {p} $- approximation theorems for these operators.

Subsequently, a global direct result for simultaneous approximation in the $ L _ {p} $- metric in terms of the second-order Ditzian–Totik modulus of smoothness was proved, see [a5]. For local direct results for simultaneous approximation of functions with polynomial growth, see [a5].

References

[a1] P.N. Agrawal, H.S. Kasana, "On simultaneous approximation by Szász–Mirakian operators" Bull. Inst. Math. Acad. Sinica , 22 (1994) pp. 181–188
[a2] V.A. Baskakov, "An example of a sequence of linear positive operators in the space of continuous functions" Dokl. Akad. Nauk SSSR , 113 (1957) pp. 249–251 (In Russian)
[a3] H. Berens, "Pointwise saturation of positive operators" J. Approx. Th. , 6 (1972) pp. 135–146
[a4] M. Heilmann, "Approximation auf durch das Verfahren der Operatoren vom Baskakov–Durrmeyer Typ" , Univ. Dortmund (1987) (Dissertation)
[a5] M. Heilmann, M.W. Müller, "On simultaneous approximation by the method of Baskakov–Durrmeyer operators" Numer. Funct. Anal. Optim. , 10 (1989) pp. 127–138
[a6] C.P. May, "Saturation and inverse theorems for combinations of a class of exponential-type operators" Canad. J. Math. , 28 (1976) pp. 1224–1250
[a7] S.P. Singh, "On Baskakov-type operators" Comment. Math. Univ. St. Pauli, , 31 (1982) pp. 137–142
[a8] Y. Suzuki, "Saturation of local approximation by linear positive operators of Bernstein type" Tôhoku Math. J. , 19 (1967) pp. 429–453
How to Cite This Entry:
Baskakov operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baskakov_operators&oldid=45993
This article was adapted from an original article by P.N. Agrawal (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article