Reading limited weekly data as time series in R I have weekly sales data for 3 months(Oct - Dec) of 2015 and I want to estimate the weekly sales of 2016 for the next three months using ARIMA model in R.
 
When I tried converting this into time series in R using - 
training_data <- ts(training_data,start = c(2015,40),end = c(2015,52),frequency = 52)
I got the result as -

How could I preserve the dates and the corresponding data I had earlier when converting to time series?
 A: When using the ts function in R, you don't need to provide the dates at all, just provide the right side column as a vector and then specify the period. 
Your post mentions 3 months of data (Oct - Dec) of 2014, but the table you're showing is for Oct 2015 - Jan 2016 (4 months). 
The command you are using is specifying Week 40 of 2015 through Week 52 of 2015,  so you are providing a table which has 18 values but trying to fit into a time series that has 12 values.
Finally - trying to predict 3 months of weekly sales data using 3~4 months of sales data is pretty much useless. 
If the data is seasonal (likely) then you can't capture any seasonality with 3 months, you will need at least 2 years to capture that. The sales patterns of most products for Jan, Feb, Mar, will be completely different from the sales patterns for Oct, Nov, March. 
Even in the unlikely scenario that your sales data doesn't have any seasonal pattern, your still stretching it very much by trying to predict 3 months using 3 months. 
The best forecast you might be able to get out of that is a moving average or simple exponential smoothing forecast for one or two weeks ahead, but as I said, that is very unlikely since sales data usually has very strong seasonal patterns. 
