<PAGE>

                                                                    EXHIBIT 31.1

    CERTIFICATION OF CHIEF EXECUTIVE OFFICER PURSUANT TO RULE 13a-14(a)/RULE
          15d-14(a) OF THE SECURITIES EXCHANGE ACT OF 1934, AS AMENDED

I, David C. Mahoney, certify that:

      1.    I have reviewed this Annual Report on Form 10-K of Applix, Inc.;

      2.    Based on my knowledge, this report does not contain any untrue
            statement of a material fact or omit to state a material fact
            necessary to make the statements made, in light of the circumstances
            under which such statements were made, not misleading with respect
            to the period covered by this report;

      3.    Based on my knowledge, the financial statements, and other financial
            information included in this report, fairly present in all material
            respects the financial condition, results of operations and cash
            flows of the registrant as of, and for, the periods presented in
            this report;

      4.    The registrant's other certifying officer and I are responsible for
            establishing and maintaining disclosure controls and procedures (as
            defined in Exchange Act Rules 13a-15(e) and 15d-15(e)) for the
            registrant and have:

                  a)    Designed such disclosure controls and procedures, or
                        caused such disclosure controls and procedures to be
                        designed under our supervision, to ensure that material
                        information relating to the registrant, including its
                        consolidated subsidiaries, is made known to us by others
                        within those entities, particularly during the period in
                        which this report is being prepared;

                  b)    [Not applicable.]

                  c)    Evaluated the effectiveness of the registrant's
                        disclosure controls and procedures and presented in this
                        report our conclusions about the effectiveness of the
                        disclosure controls and procedures, as of the end of the
                        period covered by this report based on such evaluation;
                        and

                  d)    Disclosed in this report any change in the registrant's
                        internal control over financial reporting that occurred
                        during the registrant's most recent fiscal quarter (the
                        registrant's fourth fiscal quarter in the case of an
                        annual report) that has materially affected, or is
                        reasonably likely to materially affect, the registrant's
                        internal control over financial reporting; and

      5.    The registrant's other certifying officer and I have disclosed,
            based on our most recent evaluation of internal control over
            financial reporting, to the registrant's auditors and the audit
            committee of the registrant's board of directors (or persons
            performing the equivalent functions):

                  a)    All significant deficiencies and material weaknesses in
                        the design or operation of internal control over
                        financial reporting which are reasonably likely to
                        adversely affect the registrant's ability to record,
                        process, summarize and report financial information; and

                  b)    Any fraud, whether or not material, that involves
                        management or other employees who have a significant
                        role in the registrant's internal control over financial
                        reporting.

                                        /s/ David C. Mahoney
                                        ---------------------------------------
Date: March 31, 2005                    David C. Mahoney
                                        President and Chief Executive Officer