在Apache Calcite中，用户可以通过编写自定义聚合函数来扩展其功能。以下是定义一个简单的自定义聚合函数的示例：

1. 确保您已经安装了Apache Calcite并且可以使用其Java API。

2. 编写一个类，该类实现org.apache.calcite.schema.AggregateFunction接口。例如，以下是一个计算平均值的自定义聚合函数：

import org.apache.calcite.schema.AggregateFunction;

public class AvgAggFunction implements AggregateFunction {
  private double sum = 0;
  private int count = 0;

  @Override
  public void init() {
    sum = 0;
    count = 0;
  }

  @Override
  public void accumulate(Object o) {
    if (o instanceof Number) {
      sum += ((Number) o).doubleValue();
      count++;
    }
  }

  @Override
  public Object getResult() {
    return count == 0 ? null : sum / count;
  }

  @Override
  public void merge(AggregateFunction other) {
    if (other instanceof AvgAggFunction) {
      AvgAggFunction o = (AvgAggFunction) other;
      sum += o.sum;
      count += o.count;
    }
  }
}

3. 使用org.apache.calcite.tools.RelBuilder或SqlBuilder等类以及定义的自定义聚合函数来构建SQL查询。例如：

import org.apache.calcite.tools.RelBuilder;
import org.apache.calcite.rel.RelNode;
import org.apache.calcite.sql.SqlAggFunction;
import org.apache.calcite.sql.fun.SqlStdOperatorTable;

public class ExampleQuery {
  public static void main(String[] args) {
    // Create a RelBuilder
    RelBuilder builder = RelBuilder.create();

    // Define the input table
    builder.scan("my_schema", "my_table");

    // Define the aggregation using the custom function
    SqlAggFunction avgAgg = new AvgAggFunction();
    builder.aggregate(builder.groupKey(builder.field("id")), 
                      builder.aggregateCall(avgAgg, builder.field("value").as("avg_value")));

    // Generate SQL and execute it