As voters went to the polls in a closely watched special election in the heavily Democratic House district last represented by Anthony D. Weiner, the Republican candidate expressed confidence that an upset was within reach, while his Democratic opponent played down the anxiety in his party and predicted that a strong turnout effort would propel him to victory.

A victory on Tuesday by the Republican, Bob Turner, in a traditional Democratic stronghold like New York State’s Ninth Congressional District would reinforce concern among Democrats that President Obama’s standing with voters may have deteriorated substantially, and Mr. Turner urged voters to send just such a message.

The Democratic candidate, Assemblyman David I. Weprin, who trailed by six points in a poll released on Friday, shook off questions about his campaign and said he expected to be celebrating a “big victory” on Tuesday evening.

“The only poll that counts is the one tomorrow,” Mr. Weprin said as he campaigned on Monday. “We’re going to have every union behind us, we’re going to have more troops in the field, and we’re going to pull out the traditional Democratic vote.”