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A note on the Analogue of Lebesgue-Radon-Nikodym theorem with respect to weighted p-adic q-measure on Z p

Journal of Inequalities and Applications20132013:15

https://doi.org/10.1186/1029-242X-2013-15

  • Received: 2 September 2012
  • Accepted: 13 December 2012
  • Published:

Abstract

We give the analogue of the Lebesgue-Radon-Nikodym theorem with respect to a weighted p-adic q-measure on Z p . In a special case, when the weight q x is 1, we can derive the same result as Kim et al. (Abstr. Appl. Anal. 2011:637634, 2011). And if q = 1 , we have the same result as Kim (Russ. J. Math. Phys. 19:193-196, 2012).

MSC:11B68, 11S80.

Keywords

  • p-adic q-measure
  • Lebesgue-Radon-Nikodym

1 Introduction

Let p be a fixed odd prime number. Throughout this paper, the symbols Z p , Q p , and C p denote the ring of p-adic integers, the field of p-adic rational numbers, and the p-adic completion of the algebraic closure of Q p , respectively. Let ν p be the normalized exponential valuation of C p with | p | = p ν p ( p ) = 1 p and ν p ( 0 ) = .

When one speaks of q-extension, q can be regarded as an indeterminate, a complex q C , or a p-adic number q C p . In this paper, we assume that q C p with | 1 q | < 1 , and we use the notations of q-numbers as follows:
[ x ] q = [ x : q ] = 1 q x 1 q and [ x ] q = 1 ( q ) x 1 + q .
(1.1)
For any positive integer N, let
a + p N Z p = { x Z p | x a ( mod p N ) } ,
(1.2)

where a Z satisfies the condition 0 a < p N (see [18]).

It is known that the fermionic p-adic q-measure on Z p is given by Kim as follows:
μ q ( a + p N Z p ) = ( q ) a [ p N ] q = 1 + q 1 + q p N ( q ) a , (see [1, 6, 9–12]) .
(1.3)
Let C ( Z p ) be the space of continuous functions on Z p . From (1.3), the fermionic p-adic q-integral on Z p is defined by Kim as follows:
I q ( f ) = Z p f ( x ) d μ q ( x ) = lim N 1 [ p N ] q x = 0 p N 1 f ( x ) ( q ) x ,
(1.4)

where f C ( Z p ) (see [1, 6, 912]).

Let us assume q C p with | q 1 | < 1 . By (1.4), we get
Z p q x e [ x ] q t d μ q ( x ) = n = 0 E n , q t n n ! ,
(1.5)
(see [7, 8, 13]) where E n , q are q-Euler numbers. The q-Euler polynomials, E n , q ( x ) , are also defined by
Z p q y e [ x + y ] q t d μ y ( t ) = n = 0 E n , q ( x ) t n n ! .
(1.6)
By (1.5) and (1.6), we get
E n , q ( x ) = l = 0 n ( n l ) x n l E l , q = ( x + E q ) n ,
with the usual convention of replacing ( E q ) n by E n , q (see [1, 2, 7, 8, 13]),
E n , q = Z p q x [ x ] q n d μ q ( x ) = [ ι ] q ( 1 q ) n l = 0 n ( 1 ) l ( n l ) 1 [ l ] q .

We will give the analogue of the Lebesgue-Radon-Nikodym theorem with respect to a weighted p-adic q-measure on Z p . In a special case, when the weight q x is 1, we can derive the same result as Kim et al. [9]. And if q = 1 , we have the same result as Kim [4].

2 Lebesgue-Radon-Nikody-type theorem with respect to a weighted p-adic q-measure on Z p

For any positive integer a and n, with a < p n and f C ( Z p ) , let us define
μ ˜ f , q ( a + p n Z p ) = a + p n Z p q x f ( x ) d μ q ( x ) ,
(2.1)

where the integral is the fermionic p-adic q-integral on Z p .

From (1.3), (1.4), and (2.1), we note that
(2.2)
By (2.2), we get
μ ˜ f , q ( a + p n Z p ) = [ 2 ] q [ 2 ] q p n ( 1 ) a Z p q p n x f ( a + p n x ) d μ q p n ( x ) .
(2.3)

Therefore, by (2.3), we obtain the following theorem.

Theorem 1 For f , g C ( Z p ) , we have
μ ˜ α f + β g , q = α μ ˜ f , q + β μ ˜ g , q ,
(2.4)

where α, β are constants.

From (2.2) and (2.4), we note that
| μ ˜ f , q ( a + p n Z p ) | M f q ,
(2.5)

where f q = sup x Z p | q x f ( x ) | and M is some positive constant.

Now, we recall the definition of the strongly fermionic p-adic q-measure on Z p . If μ q satisfies the following equation:
| μ q ( a + p n Z p ) μ q ( a + p n + 1 Z p ) | δ n , q ,
(2.6)

where δ n , q 0 and n and δ n , q is independent of a, then μ q is called a weakly fermionic p-adic q-measure on Z p .

If δ n , q is replaced by C p ν p ( 1 q n ) (C is some constant), then μ q is called a strongly fermionic p-adic q-measure on Z p .

Let P ( x ) C p [ [ x ] q ] be an arbitrary q-polynomial with a i [ x ] q i . Then we see that μ P , q is a strongly fermionic p-adic q-measure on Z p . Without loss of generality, it is enough to prove the statement for P ( x ) = [ x ] q k .

Let a be an integer with 0 a < p n . Then we get
μ ˜ P , q ( a + p n Z p ) = [ 2 ] q [ 2 ] q p n ( 1 ) a lim m 1 [ p m n ] q p n i = 0 p m n 1 [ a + i p n ] q k ( 1 ) i q p n i ,
(2.7)
and
q p n i = l = 0 i ( i l ) [ p n ] q l ( q 1 ) l .
By (2.7), we easily get
μ ˜ P , q ( a + p n Z p ) [ 2 ] q [ 2 ] q p n ( 1 ) a [ a ] q k ( mod [ p n ] q ) [ 2 ] q [ 2 ] q p n ( 1 ) a P ( a ) ( mod [ p n ] q ) .
(2.8)
Let x be arbitrary in Z p with x x n ( mod p n ) and x x n + 1 ( mod p n + 1 ) , where x n and x n + 1 are positive integers such that 0 x n < p n and 0 x n + 1 < p n + 1 . Thus, by (2.8), we have
| μ ˜ P , q ( a + p n Z p ) μ ˜ P , q ( a + p n + 1 Z p ) | C p ν p ( 1 q p n ) ,
(2.9)

where C is some positive constant and n 0 .

Let
f μ ˜ P , q ( a ) = lim n μ ˜ P , q ( a + p n Z p ) .
(2.10)
Then by (2.5), (2.7), and (2.8), we get
f μ ˜ P , q ( a ) = [ 2 ] q 2 ( 1 ) a [ a ] q k = [ 2 ] q 2 ( 1 ) a P ( a ) .
(2.11)
Since f μ ˜ P , q ( x ) is continuous on Z p , it follows, for all x Z p ,
f μ ˜ P , q ( x ) = [ 2 ] q 2 ( 1 ) x P ( x ) .
(2.12)
Let g C ( Z p ) . By (2.10), (2.11), and (2.12), we get
Z p g ( x ) d μ ˜ P , q ( x ) = lim n i = 0 p n 1 g ( i ) μ ˜ P , q ( i + p n Z p ) = [ 2 ] q 2 lim n i = 0 p n 1 g ( i ) ( q ) i [ i ] q k = Z p q x g ( x ) [ x ] q k d μ q ( x ) .
(2.13)

Therefore, by (2.13), we obtain the following theorem.

Theorem 2 Let P ( x ) C p [ [ x ] q ] be an arbitrary q-polynomial with a i [ x ] q i . Then μ ˜ P , q is a strongly fermionic weighted p-adic q-measure on Z p , and for all x Z p ,
f μ ˜ P , q = ( 1 ) x [ 2 ] q 2 P ( x ) .
(2.14)
Furthermore, for any g C ( Z p ) , we have
Z p g ( x ) d μ ˜ P , q ( x ) = Z p q x g ( x ) P ( x ) d μ q ( x ) ,
(2.15)

where the second integral is a fermionic p-adic q-integral on Z p .

Let f ( x ) = n = 0 a n , q ( x n ) q be the q-Mahler expansion of a continuous function on Z p , where
( x n ) q = [ x ] q [ x 1 ] q [ x n + 1 ] q [ n ] q ! ( see [4] ) .
(2.16)

Then we note that lim n | a n , q | = 0 .

Let
f m ( x ) = i = 0 m a i , q ( x i ) q C p [ [ x ] q ] .
(2.17)
Then
f f m sup m n | a n , q | .
(2.18)
Writing f = f m + f f m , we easily get
(2.19)
From Theorem 2, we note that
| μ ˜ f f m , q ( a + p n Z p ) | f f m C 1 p ν p ( 1 q p n ) ,
(2.20)

where C 1 is some positive constant.

For m 0 , we have f = f m .

So,
| μ ˜ f m , q ( a + p n Z p ) μ ˜ f m , q ( a + p n + 1 Z p ) | C 2 p ν p ( 1 q p n ) ,
(2.21)

where C 2 is also some positive constant.

By (2.20) and (2.21), we see that
(2.22)
If we fix ϵ > 0 and fix m such that f f m ϵ , then for n 0 , we have
| f ( a ) μ ˜ f , q ( a + p n Z p ) | ϵ .
(2.23)
Hence, we have
f μ f , q ( a ) = lim n μ ˜ f , q ( a + p n Z p ) = [ 2 ] q 2 ( 1 ) a f ( a ) .
(2.24)

Let m be a sufficiently large number such that f f m p n .

Then we get
μ ˜ f , q ( a + p n Z p ) = μ ˜ f m , q ( a + p n Z p ) + μ ˜ f f m , q ( a + p n Z p ) = μ ˜ f m , q ( a + p n Z p ) = ( 1 ) a [ 2 ] q [ 2 ] q p n f ( a ) ( mod [ p n ] q ) .
(2.25)
For any g C ( Z p ) , we have
Z p g ( x ) d μ ˜ f , q ( x ) = Z p q x f ( x ) g ( x ) d μ q ( x ) .
(2.26)
Assume that f is the function from C ( Z p , C p ) to Lip ( Z p , C p ) . By the definition of μ ˜ q , we easily see that μ ˜ q is a strongly p-adic q-measure on Z p , and for n 0 ,
| f μ ˜ q ( a ) μ ˜ q ( a + p n Z p ) | C 3 p ν p ( 1 q p n ) ,
(2.27)

where C 3 is some positive constant.

If μ ˜ 1 , q is associated with strongly fermionic weighted p-adic q-measure on Z p , then we have
| μ ˜ 1 , q ( a + p n Z p ) f μ ˜ q ( a ) | C 4 p ν p ( 1 q p n ) ,
(2.28)

where n 0 and C 4 is some positive constant.

From (2.28), we get
(2.29)

where K is some positive constant.

Therefore, μ ˜ q μ ˜ 1 , q is a q-measure on Z p . Hence, we obtain the following theorem.

Theorem 3 Let μ ˜ q be a strongly fermionic weighted p-adic q-measure on Z p , and assume that the fermionic weighted Radon-Nikodym derivative f μ ˜ q on Z p is a continuous function on Z p . Suppose that μ ˜ 1 , q is the strongly fermionic weighted p-adic q-measure associated to f μ ˜ q . Then there exists a q-measure μ ˜ 2 , q on Z p such that
μ ˜ q = μ ˜ 1 , q + μ ˜ 2 , q .
(2.30)

Declarations

Acknowledgements

This research was supported by the Kyungpook National University Research Fund 2012. The authors would like to thank professor T. Kim who suggested this problem to us.

Authors’ Affiliations

(1)
Department of Medical Information Technology, Daegu Haany University, Kyungsan, 712-715, Republic of Korea
(2)
Department of Mathematics Education, Kyungpook National University, Daegu, 702-701, Republic of Korea

References

  1. Bayad A, Kim T: Identities involving values of Bernstein, q -Bernoulli, and q -Euler polynomials. Russ. J. Math. Phys. 2011, 18(2):133–143. 10.1134/S1061920811020014MATHMathSciNetView ArticleGoogle Scholar
  2. Choi J, Kim T, Kim YH: A note on the q -analogues of Euler numbers and polynomials. Honam Math. J. 2011, 33(4):529–534. 10.5831/HMJ.2011.33.4.529MATHMathSciNetView ArticleGoogle Scholar
  3. Kim T: q -Volkenborn integration. Russ. J. Math. Phys. 2002, 9(3):288–299.MATHMathSciNetGoogle Scholar
  4. Kim T: Lebesgue-Radon-Nikodym theorem with respect to fermionic p -adic invariant measure on Z p . Russ. J. Math. Phys. 2012, 19(2):193–196. 10.1134/S1061920812020057MATHMathSciNetView ArticleGoogle Scholar
  5. Kim T: Lebesgue-Radon-Nikodym theorem with respect to q -Volkenborn distribution on μ q . Appl. Math. Comput. 2007, 187(1):266–271. 10.1016/j.amc.2006.08.123MATHMathSciNetView ArticleGoogle Scholar
  6. Kim T: A note on q -Bernstein polynomials. Russ. J. Math. Phys. 2011, 18(1):73–82. 10.1134/S1061920811010080MATHMathSciNetView ArticleGoogle Scholar
  7. Kim T: Note on the Euler numbers and polynomials. Adv. Stud. Contemp. Math. 2008, 17(2):131–136.MATHMathSciNetGoogle Scholar
  8. Kim T: Some identities on the q -Euler polynomials of higher order and q -Stirling numbers by the fermionic p -adic integral on Z p . Russ. J. Math. Phys. 2009, 16(4):484–491. 10.1134/S1061920809040037MATHMathSciNetView ArticleGoogle Scholar
  9. Kim T, Dolgy DV, Lee SH, Ryoo CS: Analogue of Lebesgue-Radon-Nikodym theorem with respect to p -adic q -measure on Z p . Abstr. Appl. Anal. 2011., 2011: Article ID 637634Google Scholar
  10. Kim T, Kim SD, Park DW: On uniform differentiability and q -Mahler expansion. Adv. Stud. Contemp. Math. 2001, 4(1):35–41.MATHMathSciNetGoogle Scholar
  11. Jeong J, Rim SH: A note on the Lebesgue-Radon-Nikodym theorem with respect to weighted p -adic invariant integral on Z p .Abstr. Appl. Anal. 2012., 2012: Article ID 696720Google Scholar
  12. Kim T, Choi J, Kim H: A note on the weighted Lebesgue-Radon-Nikodym theorem with respect to p -adic invariant integral on Z p . J. Appl. Math. Inform. 2012, 30(1–2):211–217.MATHMathSciNetGoogle Scholar
  13. Kim T: New approach to q -Euler polynomials of higher order. Russ. J. Math. Phys. 2010, 17(2):218–225. 10.1134/S1061920810020068MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Pak et al.; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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