Please explain me if deity worship is authorized in Bhagavad-gita?

This question is to be considered by referring to certain concepts.  

Generally people do not accept anything unless they are given proof of the facts upon which they have to base their consideration. Of course it is a common sense that everyone wants to examine the facts and only when he is convinced about their authenticity he may accept them.
But when people demand proof they have their own concept of proof and evidence in their mind as per their logic. Most of the people demand proof in writing or many of them rely upon words of superiors and so on.

It is to be understood that the method of proving many things is not by direct reference but by our proper understanding of the concept of particular topic. For example, every liquid contains base of water. But do we find on any bottle of liquid, words or caption that ‘this liquid contains water base’. Of course not always. Then how can we understand this?
If one has properly studied chemistry he would undoubtedly accept this contention and would never demand any written caption on bottle of liquid as stated above.
Similarly people study Bhagavad-gita, but they don’t know how to take words of Krishna in appropriate manner. Because they may not study Bhagavad-gita under direction of spiritual master.

Lord Krishna states in Bhagavad-gita (9.34) ‘Always think of Me, become My devotee, pay obeisances to Me, and surely you will come to me.’

Now when one desires to remember Lord Krishna, how can one know how Krishna looks like? How can one know how to pay obeisances to Krishna?

Now these things are not possible unless one is given an opportunity to know how Krishna looks like and what is the process of paying obeisances to Krishna. For example when we have never seen our forefathers, how can we know about how they looked like? Of course when we see their photo or image. And accordingly we can know about their personality and we can bow before them to pay them honor.

Similarly, as Lord is sat-cit-anand and His body is completely transcendental, it is not possible for us to know and see Him by our imperfect senses.

Therefore, in order to give us a facility to see Him Lord Himself has instructed in Srimad-Bhagavatam that He can be perceived in His icon or image form (shaili darumayi……).
This is the concept of Deity worship. So even though the actual reference of Deity worship is found in Srimad-Bhagavatam, it’s original concept is found in Bhagavad-gita (9.34).
Only what is required is a proper understanding of the verse under guidance of a spiritual master who is the representative of Krishna.

The purpose, need and scientific aspect of Deity worship can be explained by following example;

Why do we need a clock? What is scientific aspect of a telephone or television?

The answer is that all these things enable us to develop contact with those objects which are out of reach of our senses.

We need a clock because clock gives us an opportunity to understand what is the time (time factor) irrespective whether it is a watch clock or digital clock or any old-styled clock. Unless we use clock we have no chance to understand and develop connection with the time factor.

Similarly television and telephone enable us to develop contact with those objects and personalities which are beyond our jurisdiction.

Same is the concept of Deity worship. When Deity of Krishna is installed,  Krishna gives us an opportunity to reciprocate with Him through His Deity. Accordingly, we can systematically revive and develop our lost connection or relation with Him and pay obeisances to Him.

As such Krishna shows His mercy upon us by allowing us to see Him through His Deity form. This is the scientific aspect of Deity worship and its root is found in Bhagavad-gita (9.34) as already stated above.