How to control a capacitive load using an Arduino Uno
Be careful, according the output waveform diagram in the datasheet, the output is not PWM, but instead a ~1.5kHz square wave with variable amplitude and controlled edge rates. Simply feeding PWM from the Arduino to an H-bridge doesn't appear to be sufficient.

Rather, you need to create a dynamically variable voltage supply and power the H-bridge from that. The H-bridge is driven with a constant frequency square wave and the supply voltage is varied.

First off, the DC/DC: 62Vrms is ~88Vpk. So you need a ~90V boost converter. This shouldn't be too difficult, but requires some careful design. A part such as Linear's LT8331 or similar should work.

Now that you've got an HV supply, you need to figure out how to modulate it. It doesn't sound like you need really fast response time, so it might be feasible to PWM it, and filter that to get a stable supply voltage (you are basically designing a unipolar class-D amplifier). The datasheet specifies a response time of 20ms for 0% to 90%, so you might have enough separation between modulation rate and PWM frequency to make this work. On an Arduino, you'll need to ditch the built-in libraries and configure the hardware PWM to go as fast as possible (~65kHz IIRC).

Alternatively, you could try using a boost converter with dynamic feedback. However, this wouldn't work with voltages below 5V (I don't know if that's important). A SEPIC architecture might work as well, but I'm not sure how close to 0V it could get.