51005 mjis ray

DOI: 10.15415/mjis.2016.51005

On The Generalized Divided Differences SUBHASIS RAY* AND SUBHANKAR GHOSH Department of Mathematics, Siksha Bhavan, Visva-Bharati, Shantiniketan, West Bengal, India *

Email: subhasis.ray@visva-bharati.ac.in

Received: February 06, 2016| Revised: March 19, 2016| Accepted: July 02, 2016

Published online: September 05, 2016 The Author(s) 2016. This article is published with open access at www.chitkara.edu.in/publications

Abstract Let V ⊂ ℜ be a finite set and f : ℜ → ℜ . f(V) is divided difference of f at the points of V. The m-th order Peano derivative of f ({x} ∪ V ) with respect to x is called generalized divided difference and is denoted by f m ( x,V ) .

The properties of f m ( x,V ) are studied.

Keywords: Peano derivative, Divided difference, Riemann* derivative, n-convex function.

MSC 2010: Primary 26A24, 26A99 1. INTRODUCTION Let f m ( x,V ) be m-th order Peano derivative of f(x0, x1"…" xn, x) regarded as a function of x. In [1] the authors have termed the iterated limit of divided difference by generalized divided difference and use it to study the properties of n-convex functions. In this article we have studied the properties of that generalized divided difference, which is equivalent to f m ( x,V ) .

Let f : E → ℜ, V = {x0 , x1 ,…, xn } ⊂ E , then the divided difference of f at V is defined by

n

f (V ) = f ( x0 ,…, xn ) = ∑ i =0

where

n

ω( x) = ∏ ( x - xi ). i =0

f ( xi )

ω ′( xi ) Mathematical Journal of Interdisciplinary Sciences Vol-5, No-1, September 2016 pp. 53–59

Ray, S Ghosh, S

We write f(x; V) instead of f (V ∪ {x}) .

Let f : E → ℜ , and n ∈  . Then f is n-convex in E if for each subset V of E containing n+1 points, f (V ) ≥ 0 .

Let x ∈ E - V be right hand limit point of E, then right Peano derivative of divided differences with respect to the set E is defined inductively as

f1+ ( x;V ) = lim+ y→ x y∈E

f ( y;V ) - f ( x;V ) ( y - x)

and if f r+ ( x;V ) exist for 1≤ r < m , then m-th order derivative

f m+ ( x;V ) = lim+ γ m ( f , x, y,V ) y→ x y∈E

where

γ m ( f , x, y ,V ) =

m-1  m!  ( y - x )i + f ( y ; V ) f ( x ; V )  . ∑ i  ( y - x) m  i! i =0

Here we assume f 0+ ( x;V ) = f ( x;V ) .

If V = ϕ , then we write γ m ( f , x, y ) instead of γ m ( f , x, y, ϕ ) . In this case f m+ ( x;V ) is the usual right hand Peano derivative of f at x of order m and is denoted by f m+ ( x) . If x ∈ E - V be a left hand limit point, we define left hand Peano derivative of divided difference f m- ( x;V ) in similar way.

If x is both sided limit point and f m+ ( x;V ) and f m- ( x;V ) both exist and f i + ( x;V ) = f i - ( x;V ) for i = 1, 2,…, m, then f is said to have the m-th order Peano derivative of divided difference fm(x; V). Clearly for a fixed V ⊂ E , f m ( x;V ) is the m-th order Peano derivative of the function f(x; V), regarded as a function of x. Hence for x ∉ V , f m ( x;V ) exists if and only if fm(x) exists.

Let f : E → ℜ and V ⊂ E be finite and x ∈ E - V be a limit point of E. Let {x1, …, xk} be k distinct points in E - V. Then as in [1]

[ f , x,V ]k = lim ... lim f ( x, x1 ,…, xk ;V ). xk → x xk ∈ E

x1 → x x1 ∈ E

If V = ϕ , then [f, x]k is called k-th order Riemann* derivative of f at x . In what follows from now we shall drop y ∈ E under the limit notation.

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2. PROPERTIES OF f n ( x;V ) Theorem 2.1. Let f : E → ℜ be continuous, V ⊂ E be finite set and x ∈ E - V be right limit point of E. Now lim ,…, lim+ n ! f ( x0 , x1 ,…, xn ;V )

xn → x+

x0 → x

exists if and only if f m+ ( x;V ) exists and they are equal.

Proof. We prove this by induction, for n =1

lim lim f ( x0 , x1 ;V ) = lim+ f ( x, x1 ;V )

x1 → x+ x0 → x+

x1 → x

f ( x;V ) - f ( x1 ;V ) x - x1 + = f1 ( x;V ).

= lim+ x1 → x

Suppose the theorem is true for n = k, k ≥1. We prove the theorem for n = k + 1 and so the proof will be completed by induction.

So lim … lim+ r ! f ( x0 , x1 ,…, xr ;V )

xr → x+

x0 → x

and f r+ ( x;V ) exists with equal value for r = 1, 2,…, k .

Now, + lim … lim ( r + 1)! f ( x0 , x1 ,…, xr +1 ;V ) = lim ( r + 1) f r ( x;V ∪ {xr +1 }) (1) xr +1 → x

+

x0 → x

+

xr +1 → x

+

As in Newton divided difference interpolation formula, we can write

f (u;V ) = f ( x0 ;V ) + (u - x0 ) f ( x0 , x1 ;V ) + ... +(u - x0 )... (u - xr -1 ) f ( x0 , ..., xr ;V ) +(u - x0 )...(u - xr ) f (u , x0 ,..., xr ;V ).

Now performing the limit lim+ ... lim+ on both sides we get xr → x

x0 → x

f (u;V ) = f ( x;V ) + (u - x) f1+ ( x;V ) + ...

(u - x) r + (u - x) r +1 + + f r ( x;V ∪ {u}). f r ( x;V ) + r! r!

So

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On The Generalized Divided Differences

Ray, S Ghosh, S

i

(r + 1) f ( x;V ∪ {u}) = + r

f (u;V ) - f + ( x;V ) - ∑ ir=1

(u - x)

(u - x) r +1 (r + 1)!

i!

f i + ( x;V )

Putting u = xr +1 and using (1) the proof follows Remark 2.2. Theorem 2.1 also hold for left hand derivative. And for V = ϕ , it gives the well known result that Peano derivative and Riemann* derivative are equivalent. ( n ,+ ) Remark 2.3. By Theorem 2.1 f n+ ( x;V ) is same as [ f , x,V ] in [1]

Theorem 2.4. Suppose f : E → ℜ and V ⊂ E be finite set and x ∈ E - V .

Then f k ( x;V ) is divided difference of f k ( x;{t}) as a function of t. (n) Proof. As we define f n ( x;V ) is same as [ f , x,V ] of [1]. So by Corollary 3.2 of [1] the result follows.

Lemma 2.5. If f k ( x;V ) exists and if w ∈ V . Then 1

f k ( x;V ) = {kf k -1 ( x;V ) - f k ( x;V - {w})}. w- x

Proof. We prove this by induction, for k =1 f1 ( x;V ) = lim f ( x, y;V ) y→ x

f ( x;V ) - f ( x, y;V - {w}) w- x f ( x;V ) - f1 ( x;V - {w}) = w- x

= lim y→ x

So suppose, f i ( x;V ) =

1 {if i-1 ( x;V ) - f i ( x;V - {w})} for i = 1, 2,…, k - 1 w- x

Now, k -1

f k ( x;V ) = lim

f ( y;V ) - f ( x;V ) - ∑ i =1

( y - x) k k! k -1 ( y - x)i if i-1 ( x;V ) - f i ( x;V - {w}) ( y - x ) f ( x , y ;V ) - ∑ w- x i! i =1 = lim y→ x ( y - x) k k! y→ x

( y - x )i f i ( x;V ) i!

56

57

i

k!

k

(i - 1)!

1 lim w - x y→ x 1 [kf k -1 ( x;V ) - f k ( x;V - {w})] w- x

-

=

k -1

i =1

i =1

∑ i!

i

w- x

( y - x ) fi ( x; V - { x})

( y - x )i f i ( x;V - {w}) (i )!

+

k -1

n

i =0

f1 ( x0 , x1 ,…, xn ;V - {x0 , x1 ,…, xn }) = ∑ f1 ( xi ;V - {xi })

Proof. We prove it by induction. Suppose n=1 and by Lemma 2.5 we get

Theorem 2.6. Let V be a finite set and xi ∈ V for i = 0,1,…, n. . Suppose f : E → ℜ be such that f1 exists on E. Then divided difference of f1 ( x;V - {x0 ,..., xn }) at the points x0 , x1 ,…, xn , is

( y - x) k!

k

f ( y;V - {w}) - f ( x;V - {w}) - ∑

k lim w - x y→ x

=

w- x

( y - x ) f i -1 ( x ; V )

( y - x)

i =1

∑

k -1

( y - x)i-1 f i-1 ( x;V ) (i - 1)! i =1 ( y - x) k -1 (k - 1)!

k -1

f ( y ;V ) - ∑

{ f ( y ; V ) - f ( y , x; V - {w})} -

The proof is complete by induction.

y→x

= lim

(w - x)

( y - x)

On The Generalized Divided Differences

Ray, S Ghosh, S

f1 ( x0 ;V - {x0 }) =

1 { f ( x0 ;V - {x0 }) - f1 ( x0 ;V - {x0 , x1})} x1 - x0

f1 ( x1 ;V - {x1}) =

1 { f ( x1 ;V - {x1}) - f1 ( x1 ;V - {x0 , x1})} x0 - x1

Since f ( x0 ;V - { x0 }) = f ( x1 ;V - { x1 }) f1 ( x0 ;V - {x0 }) + f1 ( x1 ;V - {x1}) = f1 ( x0 ;V - {x0 , x1}) - f1 ( x1 ;V - {x0 , x1}) x0 - x1 = f1 ( x0 , x1 ;V - {x0 , x1}) So suppose it is true for (n - 1), n ≥ 2 . Hence

f1 ( x0 , x1 …  , xn ; V - { x0 , x1 , … , xn }) f ( x , … , xn-1 ; V - { x0 , x1 , … , xn }) - f1 ( x1 , … , xn ; V - { x0 , x1 , … , xn }) = 1 0 x0 - x n n-1 n ∑ i=0 f1 ( xi ; V - { xi , xn }) - ∑ i=1 f1 ( xi ; V - { xi , x0 }) = x0 - x n f1 ( x0 ; V - { x0 , xn }) - f1 ( xn ; V - { x0 , xn }) = x0 - x n n-1 ∑ i=1 { f1 ( xi ;V -{xi , xn }) - f1 ( xi ;V -{xi , x0 })} + x0 - x n

n -1

= f1 ( x0 , xn ; V - { x0 , xn }) + ∑ f1 ( xi ; V - { xi }) n

i =1

= ∑ f1 ( xi ; V - { xi }). i=0

Corollary 2.7. If f is (n + 1) convex on E, f1 exists on E; then f1 is n convex on E. Proof. Suppose f is (n+1) convex then for any finite set V = {x0 , x1 ,…, xn } ⊂ E we have f ( y, x1 ,…, xn ) - f ( x0 , x1 ,…, xn ) ( y - xo ) = lim f ( y, x0 , x1 ,…, xn ) ≥ 0

f1 ( x0 ;{x1 ,…, xn }) = lim

y → x0

y → x0

58

Similarly, f1 ( xi ;V - {xi }) ≥ 0 for all i = 1, 2,..., n.

Hence from Theorem 2.6

f1 ( x0 , x1 ,…, xn ) ≥ 0.

Therfore f1 is n convex on E. REFERENCES [1] Fejzic H, Svetic R E and Weil C E,: Differentiation of n-convex functions. Fund.

Math 209 Vol-1 9–25 (2010) zbl 1202.26008.

[2] S. N. Mukhopadhayay and S. Ray,: Mean value Theorems for divided differences and approximate Peano derivatives. Mathematica Bohemica 134 Vol-2 (2009), 165–171, zbl 1212.26075.

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On The Generalized Divided Differences