Development of linear positive operators brings major contribution in the field of approximation theory. Several mathematicians      have worked on hybrid linear positive operators. They improved rate of convergence by taking their linear combination. Here in this paper we consider a sequence of hybrid operators, combination of Beta and Baskakov basis functions,
is Beta function.
Clearly, the above operators are linear positive operators and reproduce only constant functions.
These operators can be used to approximate Lebesgue integrable functions and can be
The order of approximation for these operators is at its best at
. We can improve the order of approximation by taking linear combination of these operators.
arbitrary but fixed distinct positive integers. Then linear combination
is defined by,
is Vandermonde determinant obtained by replacing the operator column of above determinant by entries 1 given by
Simplification of (3) leads to,
. Also let
denote integral part of
. Then for sufficiently small
, the steklov mean
of m-th order corresponding to f is defined as,
We will use the following results:
has derivatives up to order m,
exists a.e and belongs to
’s are certain constants independent of f and η.
The present chapter deals with inverse and saturation results for these operators using linear approximating methods.
Operators (1) can be written as,
where the kernel,
2. Some Auxiliary Results
Here, we will present some definitions, results and lemmas which we will be needing in our main theorems.
Jensen’s Inequality. It generalizes the statement that the secant line of a convex function lies above the graph of the function. The secant line consists of weighted means of the convex function (for
Fubini’s Theorem. It gives conditions under which it is possible to compute a double integral by usng iterated integral. Order of integration may be switched if the double integral yields a finite answer when the integrand is replaced by its absolute value.
Lemma 2.3.  There exists polynomials
Lemma 2.4. For
, the mth order central moment is defined by,
1) For each
, we have,
5) For n sufficiently large and
We have the following recurrence relation,
Hence the result (8).
Lemma 2.5. For sufficiently large n and certain polynomials
in x of degree p/2 there holds,
Proof. From the above lemma 2.4 we have,
’s are certain polynomials in x of degree at most i.
Now using lemma 2.4 we will have the required result.
Lemma 2.6.   Let for
, we have,
has a compact support and K is some constant independent of n and q.
Lemma 2.7.   Let for
, we have
Here, H is some constant independent of n and q.
Lemma 2.8.   Let
Here, I is some constant independent of n and q.
Lemma 2.9.    Let
Here, J is some constant independent of n and q.
Lemma 2.10.     Let
have a compact support, there holds for
is a polynomial in x of degree j and does not vanish for all
3. Inverse Theorem
Theorem 3.1. Let
for any function
Now solving 2nd term using Jensen’s inequality and Fubini’s theorem we have
Applying (6), (11), (12) we have
For proving our theorem, we will be using mathematical induction:
Step 1: For
Now we will prove for
Step 2: For
Here, we have,
as a characteristic function of
Due to smoothness of f we have,
Using lemma 2.4, we have for
Now, using direct theorem in 
Using lemma 2.4 and interchanging integration we have
We have for
Combining all the results
Thus we have the theorem.
4. Saturation Result
Theorem 4.1. For
, then, f coincides almost everywhere with a function F on
th derivative such that,
, then for
, we have,
is arbitrarily small, so we have,
Case 1: for
Consider a sequence
and a function
such that for every
, we have from Alaoglu’s theorem,
By lemma 2.10, we have
Comparing (14) and (15) we have
We have (i) and (ii)
Case 2: for
Proceeding in the similar manner as above, we get (iii) and (iv).
Hence the theorem.
We have improved order of approximation by taking suitable linear combinations. Inverse and saturation results have been developed for our hybrid operators.
The author is grateful to the referee for many suggestions that have improved this paper a lot.