First use formatto convert your dates to the number of weeks, then plyr::ddply()to calculate the summaries:
library(plyr)
df$week <- format(df$date, format="%Y-%U")
ddply(df, .(week), summarize, income=sum(income))
week income
1 2011-52 413
2 2012-01 435
3 2012-02 379
format.date . ?strptime, , %U .
EDIT:
, , 7, , . (, , , , 1970-01-01.
:
df$week <- as.Date("1970-01-01")+7*trunc(as.numeric(df$date)/(3600*24*7))
library(plyr)
ddply(df, .(week), summarize, income=sum(income))
week income
1 2010-12-23 298
2 2010-12-30 392
3 2011-01-06 294
4 2011-01-13 152
, . .